Let be a differentiable function satisfying with initial condition

MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations

Problem

Let

y

be a differentiable function satisfying

\dfrac{dy}{dx}=y\sin x

with initial condition

y(0)=1

. A student generates a sequence of points

R_0,R_1,\dots

beginning at

R_0=(0,1)

by repeatedly moving

\Delta x=0.50

unit to the right and then adjusting the

y

–coordinate by

\Delta y=(\tfrac{dy}{dx})_{(x,y)}\,\Delta x

computed at the current point. After three steps the student reaches

x=1.50

. (a) Produce a table that shows, for each step, the current point

(x,y)

, the slope

\dfrac{dy}{dx}

evaluated there, and the resulting

\Delta y

. (b) Using the table from part (a), report the approximate value of

y

when

x=1.50

. (c) The student repeats the procedure with

\Delta x=0.25

for the interval

0\le x\le1.50

. Without recomputing every step, explain why this second approximation should be expected to differ from the first, and state whether it will likely be larger or smaller. Provide reasoning in terms of the sign and variation of

\sin x

on the interval. (d) The exact solution satisfying

y(0)=1

y=e^{1-\cos x}

. Calculate the percent error of the two approximations at

x=1.50

, and discuss which step size yields better accuracy. Show all work leading to your conclusions.

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