CPM Homework Help : APCALC Problem 8-118

A particle is moving along the $y$ -axis with a position function $y(t) =\frac { 1 } { 2 }t^2 + 2t + 1$ . Determine all times when the particle is traveling at its average rate between $0$ and $8$ seconds.

Hint:

You are being asked to find all times that the actual velocity is the same as the average velocity on the interval $[0, 8]$ . You are given the position function. How will you find the actual velocity function? How will you find the average velocity on $[0, 8]$ ?

Steps:

AROC = IROC somewhere

$\frac{y(8)-y(0)}{8-0}=v(c)$

Solve for $c$ .

The Mean Value Theorem

The Mean Value Theorem for Integrals

If $f$ is continuous on $[a, b]$ , then there exists at least one point $x = c$ in $\left(a, b\right)$ such that $\frac { 1 } { b - a } \int _ { a } ^ { b } f ( x ) d x= f(c)$ .

First quadrant, bell curve labeled, f of x, left end point on the y axis, labeled, a, right end point labeled, b, dashed horizontal segment, about 1 fourth up from x axis to peak, labeled average, & shaded rectangle between, A & b, segment & x axis, & circled points where the curve intersects the dashed segment, corresponding to tick marks on x axis, labeled, c subscript 1 & c subscript 2.

The Mean Value Theorem for Derivatives

If $F$ is continuous on $[a, b]$ and differentiable on $\left(a, b\right)$ , then there exists at least one point $x = c$ in $\left(a, b\right)$ such that $F ^ { \prime } ( c ) = \frac { F ( b ) - F ( a ) } { b - a } = f ( c )$ .