{"id":12468,"date":"2018-08-31t12:59:23","date_gmt":"2018-08-31t19:59:23","guid":{"rendered":"\/\/www.catharsisit.com\/hs\/?p=12468"},"modified":"2018-08-30t21:59:39","modified_gmt":"2018-08-31t04:59:39","slug":"interpreting-slope-fields-ap-calculus-exam-review","status":"publish","type":"post","link":"\/\/www.catharsisit.com\/hs\/ap\/interpreting-slope-fields-ap-calculus-exam-review\/","title":{"rendered":"interpreting slope fields: ap calculus exam review"},"content":{"rendered":"

slope fields show up on both the ap calculus ab and bc tests. while at first this topic might seem daunting, the questions on the test are actually quite straightforward. just keep one thing in mind: go with the flow!<\/em><\/p>\n

\"wavy
\na slope field shows the direction of flow for solutions to a differential equation.<\/p>\n

what is a slope field?<\/h2>\n

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

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

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<\/strong> suggested by the directions of the segments is a solution<\/strong> to the differential equation.<\/p>\n

\"flows each curve represents a particular solution to a differential equation.<\/p>\n

example — building a slope field<\/h3>\n

consider the differential equation dy<\/em>\/dx<\/em> = x<\/em> – y<\/em>. 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.<\/p>\n

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.<\/p>\n

now plug in each sample point (x<\/em>, y<\/em>) into the (multivariable) function x<\/em> – y<\/em>. we will keep track of the work in a table.<\/p>\n\n\n\n\n\n\n\n\n\n
x<\/em> – y<\/em><\/th>\nx<\/em> = -2<\/th>\nx<\/em> = -1<\/th>\nx<\/em> = 0<\/th>\nx<\/em> = 1<\/th>\nx<\/em> = 2<\/th>\n<\/tr>\n<\/thead>\n
y<\/em> = 2<\/td>\n-2 – 2 = -4<\/td>\n-1 – 2 = -3<\/td>\n0 – 2 = -2<\/td>\n1 – 2 = -1<\/td>\n2 – 2 = 0<\/td>\n<\/tr>\n
y<\/em> = 1<\/td>\n-2 – 1 = -3<\/td>\n-1 – 1 = -2<\/td>\n0 – 1 = -1<\/td>\n1 – 1 = 0<\/td>\n2 – 1 = 1<\/td>\n<\/tr>\n
y<\/em> = 0<\/td>\n-2 – 0 = -2<\/td>\n-1 – 0 = -1<\/td>\n0 – 0 = 0<\/td>\n1 – 0 = 1<\/td>\n2 – 0 = 2<\/td>\n<\/tr>\n
y<\/em> = -1<\/td>\n-2 – (-1) = -1<\/td>\n-1 – (-1) = 0<\/td>\n0 – (-1) = 1<\/td>\n1 – (-1) = 2<\/td>\n2 – (-1) = 3<\/td>\n<\/tr>\n
y<\/em> = -2<\/td>\n-2 – (-2) = 0<\/td>\n-1 – (-2) = 1<\/td>\n0 – (-2) = 2<\/td>\n1 – (-2) = 3<\/td>\n2 – (-2) = 4<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

<\/p>\n

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.<\/p>\n

\"slope<\/p>\n

spend some time matching each slope value from the table with its respective segment on the graph.<\/p>\n

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<\/em>\/dx<\/em> = x<\/em> – y<\/em> 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).<\/p>\n

\"slope slope field for dy<\/em>\/dx<\/em> = x<\/em> – y<\/em><\/p>\n

analyzing slope fields<\/h2>\n

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?<\/p>\n

you’ll need to master these basic skills:<\/p>\n