are you interested in pushing the boundaries of science and technology, designing video games, or innovating in the business world? if so, then rensselaer polytechnic institute might be a good fit for you!

rpi is known worldwide for a focus on science, engineering, and technology, and is consistently ranked as one of the top fifty universities in america. rensselaer’s campus boasts a business incubator, one of the most powerful university supercomputers, and an annual student video game showcase.

rensselaer polytechnic institute is also considered one of the top six universities in america in terms of median graduate income, meaning their alumni are rewarded not only with state-of-the-art facilities and rigorous academics but the tools they need to build a career straight out of college.

with a student population of 7,500 and only ten miles between campus and albany, new york, you can count on a vibrant student life in addition to a solid education.

if their motto “why not change the world?” resonates with you, rpi could be the place for you!

rensselaer polytechnic institute at a glance

some of those are pretty high numbers…but they are achievable!

let’s take a deeper dive into rensselaer polytechnic institute admissions, and find out what you need to do to make yourself a competitive candidate for rpi!

rensselaer polytechnic institute sat scores

while the composite rensselaer polytechnic institute sat scores are in the above table, here’s a breakdown by section:
 

rensselaer doesn’t have a minimum sat score required for admission, but the average composite sat score of admitted students in 2019 was a 1415 out of 1600.

on the writing test the middle 50% of students scored between a 640 and a 720, while in math the middle 50% scored between a 690 and a 780. this means 25% of admitted students scored lower than a 640 in writing or a 690 in math, while another 25% scored about a 720 in writing and a 780 in math.

if you can get scores above the middle 50% of admitted students, you can be fairly confident about your chances–if the rest of your application stands up. if not, rpi is a selective school, so you should be prepared to support your scores with strong grades and diverse extracurriculars.

rensselaer polytechnic institute act scores

the average act of students admitted in 2019 was a 31 out of 36.

the middle 50% of students, meaning those scoring between the 25% and 75% percentiles, got between a 29 and a 33. that means scoring below a 29 will put you at a disadvantage, but one quarter of students admitted that year scored lower than a 29 and still got in.

if you have a 33 or above, you can be confident about your chances: only 25% of admitted students had scores that high!

rensselaer polytechnic institute gpa average

students admitted to rpi in 2018 had an average gpa of 3.87–which means the rensselaer polytechnic institute admissions office expects you to be near the top of your class.

two-thirds of admitted freshman had a gpa of 3.75 or higher, 21% had between a 3.50 and a 3.74, 8% had between a 3.25 and a 3.49, and only 4% had gpas between a 2.00 and a 2.99.

not sure how to calculate your gpa? you can figure it out with our gpa calculator!

rensselaer polytechnic institute acceptance rate

rensselaer’s acceptance rate is 43%. this year a record total of 20,402 high school students applied to rpi, with 8,773 admitted and 1,755 enrolled.

although a 43% acceptance rate is not too selective (about 2 out of every 5 applicants are admitted!) increasing numbers of applicants may drive rpi’s admissions to become more competitive in coming years.

rensselaer polytechnic institute freshman profile

so who is the typical rensselaer freshman? let’s take a look at the class of 2022:

• international students: 13.6% from 39 countries
• students from u.s: 86.4%
• american indian/alaskan native: 0.1%
• asian: 13.7%
• black/african-american: 4.6%
• hispanic/latino: 10.4%
• multi-race (not hispanic/latino): 7.8%
• white: 61.8%
• unknown: 1.6%

rensselaer polytechnic institute other admissions requirements and info

let’s dig a little deeper, and see what else you need to get into rensselaer polytechnic institute, besides your gpa and test scores:

rpi has the following admissions recommendations:

• four years of mathematics through pre-calculus (calculus recommended)
• three years of science (including chemistry and lab-based physics)
• four years of english
• three years of social studies and/or history

additionally, the rensselaer polytechnic institute admissions committee will be looking for students who demonstrate leadership qualities and talents that will add to the rpi community!

expert advice on how to get into rensselaer polytechnic institute

we talked to ashley c. from the college prep organization transizion, to get her expert advice on how to get into rensselaer polytechnic institute university.

let’s see what she has to say!

rensselaer polytechnic institute faq

does rensselaer accept transfer students?

yes! any student with 12 or more transferable college credits completed after high school is eligible to apply to rpi as a transfer student.

transfer applications are reviewed on a rolling basis, but there are final deadlines to apply to each semester:

• fall: apply by june 1
• spring: apply by november 1
• summer (architecture only): apply by march 1

after your application is submitted, you may expect a decision in approximately 3-4 weeks!

does rensselaer accept the common app?
yes! rensselaer accepts the common app, the coalition application, or the candidate’s choice application.

the rpi admissions office weighs all applications equally, so feel free to use whichever of these application platforms is the most convenient for you!

can non-native english speakers apply to rensselaer?
yes! however, if english is not your first language, you will be required to submit your test of english as a foreign language (toefl) scores as part of your rpi applications!

how to get into rensselaer polytechnic institute

how to get into rensselaer polytechnic institute?

let’s do a quick summary!

here’s your “how to get into rensselaer polytechnic institute” checklist…

you need:

check those boxes, and you’ll be all set for your new life in new york–just don’t forget a jacket!

the post rensselaer polytechnic institute admissions: the sat, act scores and gpa you need to get in appeared first on magoosh blog | high school.

interval notation: the types of intervals

there are a few different types of intervals that commonly arise when studying math, called the open interval and the closed interval, notated respectively as (a, b) and [a, b].

interval notation for open intervals

the open interval uses parentheses, and they signify the fact that the interval contains all the real numbers x that are strictly between the numbers a and b, i.e. the interval does not actually contain the numbers a and b. another way of notating an open interval is the set of all x such that a < x < b.

interval notation for closed intervals

in the case of the closed interval, the square brackets are used to indicate that the endpoints are contained in the interval. therefore we can notate a closed interval as the set of x so that

interval notation for half-open intervals

there are slightly fancier intervals, called half-open intervals, notated as (a, b] and [a, b), which are the respective sets of all x so that , and .

an interval is called bounded when there is a real positive number m with the property that for any point x inside of the interval, we have that |x| < m.

observations on intervals

supposing as in the setup that a < b, then how many numbers are actually in the interval (a, b)? it turns out that there are uncountably infinite numbers in any interval (a, b) where a < b, no matter how close a and b are together.

it is a fact that actually, there are the same quantity of real numbers in the interval (0, 1) as there are in the entire real numbers, also represented by the interval . this seems counterintuitive, because one interval seems so much more vast than the other, but it is not a contradiction, but rather a beautiful subtly of set theory.

calculus and intervals

intervals arise regularly in calculus, and it will be important for you to know the difference between a closed interval and an open interval, since there are some theorems, like the intermediate value theorem, which requires that the interval upon which the function is defined is a closed and bounded interval.

closed and bounded intervals touch on one of the most important concepts in the broader study of calculus, that of compactness. many central theories in calculus revolve around compact sets, which in the setting of the real numbers are exactly the closed bounded intervals.

the post interval notation appeared first on magoosh blog | high school.

for a quick review of integration (or, antidifferentiation), you might want to check out the following articles first.

• ap calculus exam review: integrals
• ap calculus exam review: antidifferentiation

and now, without further ado, here are some of the most common integrals found on the ap calculus exams!

common integrals

the following seven integrals (or their close cousins) seem to pop up all the time on the ap calculus ab and bc exams.

1. remember your trig integrals!

trigonometric functions are popular on the exam!

2. simple substitutions

you need to recognize when to use the substitution u = kx, for constant k. this substitution generates a factor of 1/k because du = k dx.

for example,

3. common integration by parts

integrands of the form x f(x) often lend themselves to integration by parts (ibp).

in the following integral, let u = x and dv = sin x dx, and use ibp.

4. linear denominators

integrands of the form a/(bx + c) pop up as a result of partial fractions decomposition. (see ap calculus bc review: partial fractions). while partial fractions is a bc test topic, it’s not rare to see an integral with linear denominator showing up in the ab test as well.

the key is that substituting u = bx + c (and du = b dx) turns the integrand into a constant times 1/u. let’s see how this works in general. keep in mind that a, b, and c must be constants in order to use this rule.

5. integral of ln x

the antiderivative of f(x) = ln x is interesting. you have to use a tricky integration by parts.

let u = ln x, and dv = dx.

by the way, this trick works for other inverse functions too, such as the inverse trig functions, arcsin x, arccos x, and arctan x. for example,

6. using trig identities

for some trigonometric integrals, you have to rewrite the integrand in an equivalent way. in other words, use a trig identity before integrating. one of the most popular (and useful) techniques is the half-angle identity.

7. trigonometric substitution

it’s no secret that the ap calculus exams consist of challenging problems. perhaps the most challenging integrals are those that require a trigonometric substitution.

the table below summarizes the trigonometric substitutions.

for example, find the integral:

here, the best substitution would be x = (3/2) sin θ.

now we’re not out of the woods yet. use the half-angle identity (see point 6 above). we also get to use the double-angle identity for sine in the second line.

note, the third line may seem like it comes out of nowhere. but it’s based on the substitution and a right triangle.

if x = (3/2) sin θ, then sin θ = (2x) / 3. draw a right triangle with angle θ, opposite side 2x, and hypotenuse 3.

by the pythagorean theorem, we find the adjacent side is equal to:

that allows us to identify cos θ in the expression (adjacent over hypotenuse).

finally, θ by itself is equal to arcsin(2x/3).

the post common integrals on the ap calc exam appeared first on magoosh blog | high school.

a slope field shows the direction of flow for solutions to a differential equation.

what is a slope field?

a slope field is a visual representation of a differential equation of the form dy/dx = f(x, y). at each sample point (x, y), there is a small line segment whose slope equals the value of f(x, y).

that is, each segment on the graph is a representation of the value of dy/dx. (check out ap calculus review: differential equations for more about differential equations on the ap calculus exams.)

because each segment has slope equal to the derivative value, you can think of the segments as small pieces of tangent lines. any curve that follows the flow suggested by the directions of the segments is a solution to the differential equation.

each curve represents a particular solution to a differential equation.

example — building a slope field

consider the differential equation dy/dx = x – y. let’s sketch a slope field for this equation. although it takes some time to do it, the best way to understand what a slope field does is to construct one from scratch.

first of all, we need to decide on our sample points. for our purposes, i’m going to choose points within the window [-2, 2] × [-2, 2], and we’ll sample points in increments of 1. just keep in mind that the window could be anything, and increments are generally smaller than 1 in practice.

now plug in each sample point (x, y) into the (multivariable) function x – y. we will keep track of the work in a table.

ok, now let’s draw the slope field. remember, the values in the table above represent slopes — positive slopes mean go up; negative ones mean go down; and zero slopes are horizontal.

spend some time matching each slope value from the table with its respective segment on the graph.

it’s important to realize that this is just a sketch. a more accurate picture would result from sampling many more points. for example, here is a slope field for dy/dx = x – y generated by a computer algebra system. the viewing window is the same, but now there are 400 sample points (rather than the paltry 25 samples in the first graph).

slope field for dy/dx = x – y

analyzing slope fields

now let’s get down to the heart of the matter. what do i need to know about slope fields on the ap calculus ab or bc exams?

you’ll need to master these basic skills:

• given a slope field, select the differential equation that best matches it.
• given a slope field, estimate values of a solution with given initial condition.
• sketch a slope field on indicated sample points, from a given differential equation.

we’ve already seen above how to sketch a slope field, so let’s get some practice with the first two skills in the list instead.

sample problem 1

the slope field shown above corresponds to which of the following differential equations.

  a. dy/dx = y²

  b. dy/dx = sin y

  c. dy/dx = -sin y

  d. dy/dx = sin x

solution

look for the clues. the segments have the same slopes in any given row (left to right across the graph). therefore, since the slopes do not change with respect to x, we can assume that dy/dx is a function of y alone. that eliminates choice d.

the horizontal segments occur when y = 0, π, and -π. however, the only point at which y² equals 0 is y = 0 (not π or -π). that narrows it down to a choice between b and c.

finally, notice that slopes are positive when 0 < y < π and negative when -π < y < 0. this pattern corresponds to the values of sin y. (the signs are opposite for -sin y, ruling out choice c).

the correct choice is b.

sample problem 2

suppose y = f(x) is a particular solution to the differential equation dy/dx = x – y such that f(0) = 0. use the slope field shown earlier to estimate the value of f(2).

  a. -2.5

  b. -1.3

  c. 0.2

  d. 1.1

solution

because f(0) = 0, the solution curve must begin at (0, 0). then sketch the curve carefully following the directions of the segments. it helps to imagine that the segments are showing the currents in a river. your solution should be like a raft carried along by the currents.

then find the approximate value of f(2) on your solution curve.

the best choice from among the answers is: d. 1.1.

slope fields and euler’s method

often if you see a slope field problem in the free response section of the exam, one part of the problem might be to use euler’s method to estimate a value of a solution curve.

while the slope field itself can be used to estimate solutions, euler’s method is much more precise and does not rely on the visual representation. check out this review article for practice using the method: ap calculus bc review: euler’s method.

summary

• slope fields are visual representations of differential equations of the form dy/dx = f(x, y).
• at each sample point of a slope field, there is a segment having slope equal to the value of dy/dx.
• any curve that follows the flow suggested by the directions of the segments is a solution to the differential equation.

the post interpreting slope fields: ap calculus exam review appeared first on magoosh blog | high school.

properties of logarithms

first of all, let’s review what a logarithm is. more specifically, we need to understand how the logarithm function can be used to break down complicated expressions.

you might want to check out the following article before getting started: ap calculus review: properties of exponents and logartithms. however, the most important properties for us will be the product, quotient, and power properties for logarithms. here, we focus on a particular logarithm: the natural logarithm, ln x, though the properties remain true in any base.

in other words, logarithms change…

1. products into sums,
2. quotients into differences, and
3. exponents into multiplication.

using the properties

i like to think of the logarithm as a powerful acid that can dissolve a complicated algebraic expression.

let me illustrate the point with an example.

notice how the original expression involves a huge fraction with roots and powers all over the place. after applying the properties of logarithms, the resulting expression mostly has only plus and minus. (of course, there is a trade-off — there are now three natural logs in the simplified expression.)

logarithmic differentiation

now let’s get down to business! how can we exploit these logarithmic simplification rules to help find derivatives?

the most straightforward case is when the function already has a logarithm involved.

example 1

find the derivative of .

solution

first simplify using the properties of logarithms (see work above). then you can take the derivative of each term. but be careful — the final term requires a product rule!

the general method

in the above example, there was already a logarithm in the function. but what if we want to use logarithmic differentiation when our function has no logarithm?

suppose f(x) is a function with a lot of products, quotients, and/or powers. then you might use the method of logarithmic differentiation to find f ‘(x).

1. 1. first write y = f(x)
   2. next, apply the (natural) logarithm function to both sides of the equation.
   3. then use the product, quotient, and/or power properties to break down the expression on the right. in other words, simplify ln(f(x)) algebraically.
   4. now apply the derivative operator to both sides.
   5. use your rules of differentiation to find the derivatives. note that the left-hand side requires implicit differentiation. (check out: ap calculus review: implicit variation for details.)
      
   6. solve for dy/dx by multiplying both sides by y. in your answer, don’t forget to replace y by f(x).

example 2

use logarithmic differentiation to find the derivative of:  

solution

let’s follow the steps outlined above. the first two steps are routine.

on the other hand, step 3 requires us to break down the logarithmic expression using the properties. the work in this step depends on the function. in our case, there is a product of two factors, so we’ll start with the product property. the power property helps to break down the radical. finally, don’t forget the cancellation rule: ln(e^x) = x.

next, in steps 4 and 5 apply the derivative and work out the right-hand side.

finally, in step 6, we solve for the unknown derivative by multiplying both sides by y. don’t forget to substitute back the original function f(x) in place of y.

functions raised to a function power

a very famous question in calculus class is: what is the derivative of x^x ?

• the answer is not x x^{x – 1}, because the power rule for derivatives cannot be used when the power includes a variable.
• the answer is not x^x ln x, because the exponential rule for derivatives cannot be used when the base includes a variable.

so what is the derivative of x^x ?

well, it turns out that only logarithmic differentiation can decide this one for us!

in fact any time there is a function raised to a function power (that is, neither the exponent nor the base is constant), then you will have to use logarithms to break it down before you can take a derivative.

let’s see how it works in the simplest case: x^x.

first, write y = x^x.

then, apply the logarithm to both sides:   ln y = ln x^x.

break down the right-hand side of the equation using the algebraic properties of logarithms. in this case, only the power property plays a role.

ln y = x ln x

now you can take derivatives of the functions on both sides. but be careful… the function on the right requires a product rule.

(1/y)(dy/dx) = (1) ln x + x(1/x) = ln x + 1

finally, multiply both sides by the original function (y = x^x) to isolate dy/dx.

dy/dx = x^x(ln x + 1)

and there you have it! the derivative of x^x turns out to be trickier than you might have thought at first, but it’s not impossible.

summary

• logarithmic differentiation is a method for finding derivatives of complicated functions involving products, quotients, and/or powers.
• you can use the algebraic properties of logarithms to break down functions into simpler pieces before taking the derivative.

the post what is logarithmic differentiation? ap calc review appeared first on magoosh blog | high school.

in this review article, we’ll explore the methods and applications of linear approximation. we’ll also take a look at plenty of examples along the way to better prepare you for the ap calculus exams.

linear approximation and tangent lines

by definition the linear approximation for a function f(x) at a point x = a is simply the equation of the tangent line to the curve at that point. and that means that derivatives are key! (check out how to find the slope of a line tangent to a curve or is the derivative of a function the tangent line? for some background material.)

formula for the linear approximation

given a point x = a and a function f that is differentiable at a, the linear approximation l(x) for f at x = a is:

l(x) = f(a) + f '(a)(x – a)

the main idea behind linearization is that the function l(x) does a pretty good job approximating values of f(x), at least when x is near a.

in other words, l(x) ≈ f(x) whenever x ≈ a.

example 1 — linearizing a parabola

find the linear approximation of the parabola f(x) = x² at the point x = 1.

  a. x² + 1

  b. 2x + 1

  c. 2x – 1

  d. 2x – 2

solution

c.

note that f '(x) = 2x in this case. using the formula above with a = 1, we have:

l(x) = f(1) + f '(1)(x – 1)

l(x) = 1² + 2(1)(x – 1) = 2x – 1

follow-up: interpreting the results

clearly, the graph of the parabola f(x) = x² is not a straight line. however, near any particular point, say x = 1, the tangent line does a pretty good job following the direction of the curve.

how good is this approximation? well, at x = 1, it’s exact! l(1) = 2(1) – 1 = 1, which is the same as f(1) = 1² = 1.

but the further away you get from 1, the worse the approximation becomes.

approximating using differentials

the formula for linear approximation can also be expressed in terms of differentials. basically, a differential is a quantity that approximates a (small) change in one variable due to a (small) change in another. the differential of x is dx, and the differential of y is dy.

based upon the formula dy/dx = f '(x), we may identify:

dy = f '(x) dx

the related formula allows one to approximate near a particular fixed point:

f(x + dx) ≈ y + dy

example 2 — using differentials with limited information

suppose g(5) = 30 and g '(5) = -3. estimate the value of g(7).

  a. 24

  b. 27

  c. 28

  d. 33

solution

a.

in this example, we do not know the expression for the function g. fortunately, we don’t need to know!

first, observe that the change in x is dx = 7 – 5 = 2.

next, estimate the change in y using the differential formula.

dy = g '(x) dx = g '(5) · 2 = (-3)(2) = -6.

finally, put it all together:

g(5 + 2) ≈ y + dy = g(5) + (-6) = 30 + (-6) = 24

example 3 — using differentials to approximate a value

approximate using differentials. express your answer as a decimal rounded to the nearest hundred-thousandth.

solution

1.03333.

here, we should realize that even though the cube root of 1.1 is not easy to compute without a calculator, the cube root of 1 is trivial. so let’s use a = 1 as our basis for estimation.

consider the function . find its derivative (we’ll need it for the approximation formula).

then, using the differential, , we can estimate the required quantity.

exact change versus approximate change

sometimes we are interested in the exact change of a function’s values over some interval. suppose x changes from x₁ to x₂. then the exact change in f(x) on that interval is:

δy = f(x₂) – f(x₁)

we also use the “delta” notation for change in x. in fact, δx and dx typically mean the same thing:

δx = dx = x₂ – x₁

however, while δy measures the exact change in the function’s value, dy only estimates the change based on a derivative value.

example 4 — comparing exact and approximate values

let f(x) = cos(3x), and let l(x) be the linear approximation to f at x = π/6. which expression represents the absolute error in using l to approximate f at x = π/12?

  a. π/6 – √2/2

  b. π/4 – √2/2

  c. √2/2 – π/6

  d. √2/2 – π/4

solution

b.

absolute error is the absolute difference between the approximate and exact values, that is, e = | f(a) – l(a) |.

equivalently, e = | δy – dy |.

let’s compute dy ( = f '(x) dx ). here, the change in x is negative. dx = π/12 – π/6 = -π/12. note that by the chain rule, we obtain: f '(x) = -3 sin(3x). putting it all together,

dy = -3 sin(3 π/6 ) (-π/12) = -3 sin(π/2) (-π/12) = 3π/12 = π/4

ok, next we compute the exact change.

δy = f(π/12) – f(π/6) = cos(π/4) – cos(π/2) = √2/2

lastly, we take the absolute difference to compute the error,

e = | √2/2 – π/4 | = π/4 – √2/2.

application — finding zeros

linear approximations also serve to find zeros of functions. in fact newton’s method (see ap calculus review: newton’s method for details) is nothing more than repeated linear approximations to target on to the nearest root of the function.

the method is simple. given a function f, suppose that a zero for f is located near x = a. just linearize f at x = a, producing a linear function l(x). then the solution to l(x) = 0 should be fairly close to the true zero of the original function f.

example 5 — estimating zeros

estimate the zero of the function using a tangent line approximation at x = -1.

  a. -1.48

  b. -1.53

  c. -1.62

  d. -1.71

solution

d.

remember, the purpose of this question is to estimate the zero. first of all, the tangent line approximation is nothing more than a linearization. we’ll need to know the derivative:

then find the expression for l(x). note that g(-1) ≈ 2.13 and g '(-1) = 3.

l(x) = 2.13 + 3(x – (-1)) = 5.13 + 3x.

finally to find the zero, set l(x) = 0 and solve for x

0 = 5.13 + 3x  →  x = -5.13/3 = -1.71

the post linear approximation: ap calculus exam review appeared first on magoosh blog | high school.

how to get into university of washington? read on to learn all about university of washington sat scores, university of washington act scores, and university of washington admissions.

how to get into university of washington: an introduction to university of washington admissions

the university of washington is ranked number 58 in national universities by u.s. news and world report. this puts u washington at roughly the 97th percentile for university quality, making it very competitive. how competitive? let’s take a quick look at university of washington admissions stats, in terms of university of washington sat scores, university of washington act scores, university of washington gpa, and the school’s acceptance rate.

• university of washington sat scores (whole test, middle 50%):
  1240–1470
• university of washington sat scores (math, middle 50%):
  620–770
• university of washington sat scores (reading and writing, middle 50%):
  600-700
• university of washington act scores (whole-test score, middle 50%):
  27-33
• university of washington acceptance rate (freshman applicants):
  52%
• university of washington gpa range (freshman applicants):
  3.7-4.0

(source: us news & world report)

we’ll go over test scores and gpa in more detail, but don’t forget the other factors university of washington may look into!

university of washington sat scores: a closer look

university of washington admissions superscores your sat scores. but what the heck does this mean?

if you’re picturing an sat score report in tights and a cape, able to do things that ordinary sat score reports can’t, you’re not far off, actually. while your sat score report won’t literally be flying through the sky in heroic garb and colors, u washington allows your sat report to do something no ordinary sat report could do at most schools. sat superscoring allows your sat score report to be combined with another one, so that you get a combination of the best section scores.

if that still sounds confusing, let me give you an example. suppose you took the sat last year and got a 610 in both math and in the reading and writing section. this is frustrating because it puts your score in the middle 50% range for university of washington admissions for reading and writing, but not for math. so close and yet so far!

this is where your superscore can save the day! next, imagine you retook the sat just last month, and this time you got a 630 in math… but a 570 in reading and writing. you just can’t win… or can you? but because the university of washington accepts superscores, you can win there. you can submit both score reports, and uw will accept the highest section scores from each report. you now have a 630 in math and a 610 in reading and writing. it’s an interesting aspect of how to get into university of washington.

university of washington act scores: university of washington admissions allows superscoring here too!

act superscores are also a thing at the university of washington admissions office. university of washington admissions lets you average act section scores from different tests.

again, i’ll illustrate the magic of superscoring with an example. let’s say you get these act section scores on your first attempt: 28 english, 30 reading, 31 math, 29 science. and suppose your retake scores are: 31 english, 33 reading, 29 math, 27 science. university of washington lets you combine the best sections on both tests, as follows: 31 english, 33 reading, 31 math, 29 science.

how to get into university of washington: your odds (university of washington acceptance rate)

as i mentioned above, the university of washington’s acceptance rate is 52% for incoming freshman. however, there is far more to the university of washington acceptance rate than just the freshman numbers. let’s take a closer look.

being a washington resident greatly increases your odds of getting a coveted university of washington acceptance letter. the university of washington is a public university run by the state of washington; as you might expect, they prioritize serving state residents. in-state applicants have a 59 acceptance rate, giving them an advantage.

in contrast, being a transfer decreases your odds when it comes to how to university of washington admissions. transfer students have a 47% acceptance rate. this low acceptance rate is true even for in-state applicants who are transferring from other u washington schools.

want to increases your chances of getting into u washington? according to matt bishop, assistant director at the office of admissions, take the most challenging courses you can:

university of washington gpa and university of washington admissions

university of washington’s gpa for incoming freshmen overall is 3.7–4.0.

it’s helpful to keep track of your gpa as you complete classes, whether you’re a high school u washington hopeful or an aspiring transfer student. to figure out what your gpa will be as your grades come in, use magoosh’s gpa calculator tool.

university of washington freshman profile

some data from this section is from 2020, due to a lack of new information from the school.

one of the most striking things about the mix of washington university freshmen is just how diverse a group they are. although washington state residents make up the majority of the student body, non-residents are almost evenly divided between american students and international ones. this means that 18% of washington state’s students–nearly 1 in 5– are from other countries.

many of the most represented foreign countries on campus are what you’d expect. for example, the university of washington accepts many applicants from india, china, and south korea. but the most represented country for u washington’s international students is canada, a nation that sends relatively few international undergrads to the united states. indonesia and malaysia, two other countries that are usually under-represented in terms of international students, also send many students to study at uw.

not only that, but just about every region of the u.s. is well-represented at the university of washington, with a near equal amount of students coming from the midwest, the east, and west coasts, the south and southwest, etc. even non-mainland americans are well-represented among incoming freshman, with hawaii as one of the top sources of out-of-state students.

in short, the freshman profile at the university of washington is wonderfully diverse. no matter where you are from, if you attend the university of washington, you’re likely to meet people from home. and at the same time, you’ll get to learn and grow with learners from all over the nation and the world.

how to get into university of washington: other admissions requirements and info

we’ve already discussed gpa and the expectations for university of washington sat scores and university of washington act scores. but what about other requirements, such as letters of recommendation or admissions essays?

let’s start with letters of recommendation. the university of washington takes a strong position against letters of recommendation. university of washington admissions for undergrads will not even consider letters of recommendation. moreover, they actually ask students not to submit reference letters. given uw’s stance, a letter of recommendation could actually hurt your chances of acceptance as an undergrad.

of course, this prohibition on recommendations only applies to baccalaureate studies. university of washington admissions does ask for letters of recommendation for its grad programs. and a number of other top schools will ask for reference letters, both for graduate and undergraduate students. if you’re applying to multiple schools or already thinking of grad school, you may enjoy magoosh’s article on how to ask for a letter of recommendation for college.

now let’s talk admissions essays. these are required by university of washington admissions. in fact, your admissions essays (yes, more than one!) are a vital part of how to get into university of washington. you’ll be expected to write one longer essay (maximum of 650 words), and one “short statement” (maximum of 300 words). the good news is that these aren’t hard-to-write “research essays.” nor are the keys to a successful essay shrouded in mystery. instead, the university of washington has a web page that lists all possible essay questions, with detailed instructions. magoosh offers a secondary resource that may help as well. many aspects of our common app essay tutorial are relevant to the university of washington admissions essay.

how to get into university of washington: frequently asked questions

and now, for a roundup of frequently asked questions about university of washington admissions, university of washington sat scores, university of washington act scores, and how to get into the university of washington.

is a 1200 on the new sat good for university of washington admissions?
it would be fair to describe a 1200 on the new sat as good but not great where university of washington admissions are concerned. this would put you below the middle 50% of applicants.

what is the out of state acceptance rate for the university of washington?
out-of-state applicants have a 51% acceptance rate, compared to the overall average rate of 52%. (this lower acceptance rate for non-washingtonians applies both to students from other states and to international students from other countries.)

where can i find detailed university of washington admissions statistics?
the best place to find university of washington admissions statistics is on the university of washington website itself. an detailed list of the most recent university of washington admissions statistics can be found here.

how to get into the university of washington: conclusion

there’s a perception that public universities are less competitive than private ones–in america at least. while this can sometimes be true, the university of washington does not follow this trend. it’s a highly competitive school that attracts applicants from across the globe.

getting into the university of washington can be a challenge. but to quote dr. kelso on scrubs (a sitcom from 10 years ago, for you young college applicants), “nothing in this world that’s worth having comes easy.” be prepared to work hard to get in. but certainly don’t think of university of washington admissions as “mission: impossible” (or “admissions impossible,” to use a terrible pun). armed with the knowledge in this article, you can crack university of washington admissions, and embark on a diverse, high-quality learning experience.

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average value of functions

suppose f is a continuous function defined over an interval [a, b]. in particular f(x) exists at every one of the infinitely-many points x between and including a and b. so, if you’re looking for the average value of f on that interval, it won’t do any good to try adding up those infinitely-many data points.

instead, the way to tame the infinity is to use calculus. specifically, we define the average value of a function f as the following definite integral.

the theory behind the formula

but where does the integral formula for average come from?

the key is sampling. if you included enough of the function values, say a thousand, a million, or even more, then that should approximate the average of all infinitely-many points!

let’s illustrate with the following example.

estimate the average value of the function f(x) = √(x) + 1 over the interval [1, 3].

since we’re just estimating, let’s pick four sample points. (the more sample points you pick, the better your estimated average will be.)

divide the interval [1, 3] into four equal subintervals, and let’s agree to choose the midpoint of each subinterval. then plug those midpoints into f to find the sample values.

finally, use the familiar old averaging formula. add up the data and divide by the number of data points:

(2.12 + 2.32 + 2.50 + 2.66)/4 = 2.4

so the (approximate) average of the function is 2.4.

taking it to the limit

however, what we’ve just done will not give us an exact answer because we’ve essentially ignored most of the function! what about f(1.3) or f(2.95234)? no matter how many sample points we include, there will always be some missing… unless we can use the magic of calculus to catch them all.

the sampling process should remind you of a riemann sum. for a quick reminder, feel free to check out ap calculus review: riemann sums.

let n be any whole number and x_k^* stand for the various sample x-values. then the estimated average is the sum:

next, allow n → ∞ using a limit. we also need to get Δx into the act somehow. the trick is to multiply and divide by (b – a). remember, Δx = (b – a)/n.

examples

ok, now that you’ve seen the theory, let’s use the formula in practice!

problem 1

find the exact average value of f(x) = √(x) + 1 over the interval [1, 3].

solution

above, we only estimated the average to be 2.4. now we’ll use the integral formula to determine the average value precisely.

(it’s interesting to compare our estimate with the exact value above. √(3) + 2/3 ≈ 2.39871747, which means that our estimate of 2.4 was actually pretty good!)

problem 2

the amount of energy associated with a certain chemical reaction is given by e = x ln x, where 1 ≤ x ≤ e, and x represents the amount of one of the reactants. find the average energy of the reaction over the range of possible levels of reactant.

solution

this problem seems more like chemistry than math!

however, the keyword average tells us that mathematics plays a major role in this problem. in fact, they are simply asking for the average value of f(x) = x ln x, over the interval [1, e].

first set up the integral formula with a = 1 and b = e. then work out the integration, which involves integration by parts in this case.

thus the average energy of the reaction is (e² + 1)/[ 4(e – 1) ], or roughly 1.22.

mean value theorem for integrals

averages are also called means. so you may use the same formula to find the mean value of a function.

there is also an important result in calculus that relates the mean value to a particular function value on the given interval.

the mean value theorem for integrals (mvti). if f is continuous on a closed interval [a, b], then there is at least one point x = c in that interval such that the mean value of the function is equal to f(c). that is,

(caution: there is also a mean value theorem for derivatives. it’s important not to confuse the two.)

problem 3

let f(x) = 6x² – 8x + 1. determine the value of c at which the mean value of f on [-1, 1] is the same as f(c).

solution

according to the mean value theorem for integrals, there must be at least one such value c. let’s set up the formula and find it!

at this point, we will need to solve a quadratic equation. don’t forget your quadratic formula!

6c² – 8c – 2 = 0

c = (2 ± √(7))/3

it seems as though there may be two answers. however, only one lies within the given interval [-1, 1].

c = (2 + √(7))/3 ≈ 1.549, not in the interval.

c = (2 – √(7))/3 ≈ -0.215, in the interval.

therefore, the only value that satisfies the mvti is c = (2 – √(7))/3.

summary

although average value and the mean value theorem for integrals are specialized topics and only show up in a few problems on any given ap calculus test, they are important concepts to master. for one thing, they illustrate how integral calculus can be used in applications.

moreover, working out the average value of a function is no more difficult than computing a definite integral. so now when you see these kinds of problems on the ap calculus exam, you can rise to the challenge!

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let’s talk about how to get into penn state.

first, it’s important to distinguish between the penn state, and other campuses administered by penn state. what do i mean by this? well, penn state is actually a university system, not a single school. but the most famous penn state university is penn state-university park. this is the one nationally ranked university in the system. it’s also the penn state you are likely trying to get into if you’re reading this article. and it’s a good choice! us news and world report ranks penn state at number 57 nationally, putting the school in the top 4% for universities in the united states.

so how do you get into penn state–university park, more commonly referred to just as penn state? here are some basic stats on penn state sat scores, penn state act scores, penn state gpa, and penn state’s acceptance rate:

(sources: us news and world report, penn state admissions statistics web page.)

how to get into penn state: penn state sat scores

like most top schools, penn state does not set an absolute rule for minimum sat score. instead, it lists the sat scores of the middle 50% of accepted applicants.

but how competitive are penn state’s “middle 50” sat scores? to get a sense of that, we need to think not just in terms of scores, but also in terms of percentiles. according to the college board’s he 1250-1430 range for penn state sat scores represents a percentile range between 81st percentile and 96th. this is actually a fairly broad percentile range. the lower end of the middle 50%, a 1250/81st percentile, is actually fairly attainable. that upper 1430/96th percentile can be harder to reach. in any case, remember this: the lower your sat score is, the stronger the rest of your application will need to be. and the higher your sat score is above 1250, the better!

penn state act scores and penn state admissions

if you look at the act’s own official act to sat score conversion table, you’ll see that penn state’s 28-32 act score range is exactly equivalent to their sat 1250-1430 range. in other words, penn state act scores and penn state sat scores require you to make the same standards. so there’s no clear advantage to submitting act scores vs. sat scores to penn state admissions.

or is there? there are many factors to consider when taking the act vs. sat. so if you are trying to decide if you should take the act for penn sate, check out magoosh’s helpful article, “act vs. sat: the ultimate guide to choosing the right test.”)

and always remember– aim for the best score you can. a perfect or near-perfect score on the act can be a good thing to aspire to when applying to a competitive school like penn state.

ultimately, however, these differences are pretty small. when deciding which test is right for you, act or sat, it’s better to consider many other factors.

how to get into penn state: what does your gpa really mean for penn state admissions?

let’s take a closer look at what that 3.55 to 3.97 penn state gpa range really means. in a nutshell, it means that to have the best shot at getting into penn state, your high school grades need to be near-perfect. this is because a that penn state gpa of 3.55 is equivalent to an a- average. and a 3.97 gpa is equal to an a or an a+ average.

one thing you can take away from this is that penn state cares a lot about high school grades. it’s also tempting to take away a false message: that if you have something short of an a- average, you shouldn’t bother to apply. not true! remember, penn state sets no official minimum test scores or gpas. if your gpa is a little below 3.55, it’s probably still worth it to apply, especially if the rest of your application is strong.

if you’re not quite sure how to calculate your gpa for penn state admissions purposes, magoosh can help! check out our article on how to calculate your gpa. this should help you figure out the “gpa” part of how to get into penn state.

penn state freshman profile

we’ve already taken a good look at what a penn state freshman looks like in terms of penn state sat scores, penn state act scores, and penn state gpa. but what does a penn state freshman look like on a personal level? where do penn state students come from? where do they live? once you figure out how to get into penn state, will you fit in and thrive upon arrival?

first, let’s talk about where penn state undergraduates are from geographically. as you might expect, more than half of penn state’s students are from pennsylvania. but only a little more than half. roughly 60% of the incoming undergraduates are in-state students. approximately 30% of students are from other parts of the u.s. and around 10% of students are international. so if you attend penn state, you’ll get a strong taste of local pennsylvania culture. but you’ll also get to interact with people from many other parts of the country and the world.

now, let’s talk about where penn state freshman are form economically. to be sure, penn state has its share of wealthy and well-connected students, as you’d expect from an elite university. however, a substantial number of penn state students come form either middle-class or lower-income households. 45% of incoming freshman are found to me eligible for needs-based financial aid.

so that’s where penn state students come from, physically and financially. but once they arrive, where do they live? the vast majority of penn state freshman live in the dorms at first, since they’re actually required to live either on-campus in their first year. however, once a group of freshman move on to their second year of schooling (or later), they stop living in student housing for the most part. over all, only 35% of penn state students live on campus.

(sources for this section: penn state admissions statistics, us news and world report.)

how to get into penn state: other penn state admissions requirements and info

so we’ve discussed the penn state sat and act scores. and we’ve looked at penn state gpa. while those are among the most important penn state admissions requirements, there are a few other important things to consider.

first, you should consider penn state’s high school prerequisites. you’ll need four years of high school english. these four years must include at least 1 year of english literature and at least 1 year of english composition. and for high school math, successful penn state applicants must have taken a year each of algebra, algebra ii, and geometry. in addition, you’ll need to complete at least a half year of one of the following high school math topics: trigonometry, pre-calculus, or calculus. finally, you’ll need 3 combined years of social studies, arts, or humanities, and 2 years of a foreign language.

next, it’s important to remember that there are additional requirements for certain undergraduate majors. to see a list of special major-specific requirements for undergrad applicants to penn state, visit the official undergraduate requirements page for penn state admissions.

the penn state faq

here’s a quick faq to cover your most common additional questions students have about penn state, besides the questions i already answered above.

is penn state an ivy league school?
actually, it’s the university of pennsylvania, the other major top school in the state, that’s in the ivy league. penn state is a high ranking school but is not an elite private school like the eight members of the ivy league.

does penn state require the essay portion of the sat or act?
no, penn state does not require applicants to take the essay part of either the sat or act. however, penn state admissions is willing to look at and consider essay scores.

what is penn state best known for?
academically, penn state is best known for several different departments. the most popular majors at penn state are engineering, computer science, business, and communication/journalism.

how much does it cost to attend penn state?
for the 2019-2020 school year, penn state’s full-time tuition costs $18,454 per year for in-state students. students from out of sate are charged $34,858 per year, however. to cover additional non-tuition expenses for living and studying, penn state estimates students will also need to spend approximately $13,410 to $16,632 per year. for a detailed breakdown of penn state expenses, see the official penn state tuition and costs website for undergraduate penn state admissions.

does penn state use the common application?
yes. as of june 2018, penn state now uses the common application. the common application is a third-party website that provides the application forms for many top schools. so the common app is now an important part of how to get into penn state, and it can be important at other universities too. to learn more about the common application and its essay check out this magoosh blog post about the common app essay.

does penn state require letters of recommendation?
for most penn sate applicants, letters of recommendation are neither required nor considered. however, if you want to apply to penn state as an honors student, you will need two letters of recommendation. here again, magoosh has a helpful blog post. if you think you may need recommendations for penn state or any other school you’re applying to, we have a useful article on how to ask for a letter of recommendation for college.

how to get into penn state: conclusion

penn state is an especially good choice if you are interested in enrolling in its top majors, which are geared toward communications, business, and stem.

but penn state is really for everyone. it’s a big school with a lot to offer. if you can meet the high standards for penn state sat scores, penn state act scores, penn state gpa, and penn state admissions, penn state will have a lot to offer you personally, regardless of your chosen major. so if you’re considering penn state as an option, good! i hope this guide to how to get into penn state has been helpful.

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basic differential equations: integration

you have probably worked out hundreds of differential equations without even realizing it!

it’s true! let me give you an example.

what makes this a differential equation?

well think about what the notation means. you know that integration is the opposite of differentiation. so what we’re looking for here is a function f(x) whose derivative is equal to 2x. in other words, we have to solve:

f ‘(x) = 2x

that’s a relation involving an unknown function f and its derivative.

but this differential equation is trivial to solve! just use the power rule for integrals (or guess-and-check).

f(x) = x² + c

(for a full review of integration, check out: ap calculus exam review: antidifferentiation or ap calculus exam review: integrals.)

initial value problems

now suppose you have more information about an unknown function, such as its value at a certain point. then you may be able to solve for the function explicitly, rather than getting stuck with an unknown constant of integration at the end.

an initial value problem typically gives a derivative expression along with a function value. the goal is to produce the original function.

example 1: initial value problem

if f ‘(x) = 1/x² for x > 0, and f(1) = 5, find the expression for f(x) for x > 0.

solution

because the derivative expression is given, we integrate to find the original function. don’t forget about your constant of integration!

next, plug in the known info and solve for the unknown constant of integration.

this implies that the original function must be:    f(x) = -1/x + 6.

separation of variables

of course not every differential equation (or de) problem can be handled by simple integration. often more advanced techniques must be used, especially if both x and y show up in the equation.

for example, the following de cannot be cracked by integration (…not in its current form, that is).

the key is to somehow break up the dy and dx.

the derivative notation dy/dx looks like a fraction. however, it’s not really a fraction, because the two quantities dy and dx (called differentials) are supposed to represent the idea of taking Δx to zero in the limit. in other words, if you had to identify the numerical “value” of a differential, it would make sense to say dx = dy = 0. and we all know that 0/0 is undefined.

but those differential gadgets are more subtle than that. they really don’t carry a numerical value. instead, they stand for a limiting process. and if we’re very careful, we can work with dx and dy individually.

the method of separation of variables works by manipulating dy/dx as if it were a fraction, and then using integration to get rid of the differentials.

now let’s take another look at our example above.

example 2: separation of variables

if    , and y(0) = -3, find an equation for y in terms of x.

solution

first multiply dx to both sides. then multiply or divide as necessary so that only expressions of y are on the left, and only those with x are on the right.

next, we use the magic of calculus! all you have to do is to apply the integral symbol to each side of the equation. now you have two separate integrals to work out.

now even though both sides generated a constant of integration, those can be combined into a single constant on the right hand side. the next few steps are just for isolating y.

notice that the quantity e^c is replaced by another constant k. this is because c is still arbitrary (and unknown). so just think of k as a related, but still arbitrary constant.

finally we solve for k using the known information, y(0) = -3.

(by the way, we will drop the “±” notation at this point because k itself can take care of that choice.)

therefore, replacing k by its computed value -3, we obtain the final form of the function.

setting up differential equations

fortunately you won’t encounter any de problems on the ap calculus exam that can’t be handled by either integration or separation of variables.

however, you may be asked to set up and/or analyze a de (without solving it).

example 3: setting up a de

a certain bacteria culture has p cells at time t and grows in proportion to the square root of the amount present. set up a differential equation that models a(t).

solution

because the culture grows at a certain rate with respect to time, we know we’ll be working with the derivative da/dt.

let k be the constant of proportionality. then translate the given information into mathematics:

da/dt = k × √(a)

remember, we don’t have to solve for a(t); just set up the de that could be used to solve for it. so we’re done at this step!

example 4: analyzing a de

the number y of people infected with the flu in a certain town at time t can be modeled by the differential equation, dy/dt = 0.00002y(21500-y), where y(0) = 1. determine the limit of y as t → ∞.

solution

it would be difficult to solve this de for y explicitly. instead, look closely at the equation. it has the form of a logistics equation.

that means we can answer this question with no work at all! you just have to remember that logistics equations always have solutions that tend toward the carrying capacity m as t → ∞. here, m = 21500. that’s all you need to answer the question!

the limit will be 21500.

(for more details about the logistics equation, i recommend: ap calculus bc review: logistics growth model.)

conclusion

differential equations show up only sparingly on an ap calculus exam. but it’s important to be aware of the techniques for solving them, setting them up, and analyzing them.

• remember the basic methods of integration and separation of variables.
• know how to set up a differential equation.
• look for situations in which you may avoid solving the de.

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