CPM Homework Help : CALC3RD Problem 7-18

It just so happens that Hustling Harry’s grade in math class at any week $t$ during the semester is calculated by $g(t)=46\cos(\frac{t}{10})+10\sin(\frac{t}{2})+40$ . At what point during the $18$ -week semester is Harry’s grade equal to his average grade for the semester?

Hint:

This is an application of the Mean Value Theorem.

Partial Answer:

Show setup and steps. $t ≈7.630$ weeks

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 $F ^ { \prime } ( c ) = \frac { F ( b ) - F ( a ) } { b - a } = f ( c )$ .