A student defines F(x)=c+ _ 2 ^ x (t^ 3 -6t ) ,dt

MATHEMATICS · Multi-part · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

A student defines

F(x)=c+\int_{2}^{x}\bigl(t^{3}-6t\bigr)\,dt,

where

c

is a real constant. A second student instead writes

G(x)=\int_{5}^{x}\bigl(t^{3}-6t\bigr)\,dt,

so that

G(5)=0

. Both claim their functions model the same physical process. (a) Show that for every

x\in\mathbb R

, one can write

F(x)=G(x)+K

for a constant

K

depending only on

c

. Determine

K

explicitly in terms of

c

. (b) If the physical process is known to satisfy

F(2)=4

, determine the value of

c

, and hence state the explicit relationship between

F

and

G

. (c) Using your result from part (b), verify that

F

and

G

obey the same differential equation and explain why they represent vertically translated versions of the same curve. (d) Prove rigorously that no choice of

c

other than the one found in part (b) will satisfy the initial condition

F(2)=4

. Discuss how this reflects uniqueness of solutions for first-order initial-value problems.

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