Let $g(x)=\int_0^xf(t)dt$ where $y = f (t)$ is shown at right.

1. Evaluate $g(2)$ and $g(4)$.

   Hint (a):

   $\text{Since }g(x)\text{ is defined as }\int_{0}^{x}f(t)dt:$

   $g(2)=\int_{0}^{2}f(t)dt$

   $g(4)=\int_{0}^{4}f(t)dt$

2. Is $g ( - 2 ) = \int _ { - 2 } ^ { 0 } f ( t ) d t$?

   Hint (b):

   Consider the symmetry of $y = f(t)$ between $t = −2$ and $t = 2$.

3. Express $\int _ { - 2 } ^ { 2 } f ( t ) d t$ in terms of $g$.

   Hint (c):

   $\int_{-2}^{2}f(t)dt=kg(a)$

   What is $k$? What is $a$?

4. Is $g$ differentiable over the interval $−2 < x < 6$ ? Explain.

   Hint (d):

   Points of NON-differentiability include cusps, endpoints, jumps, holes, and vertical tangents.

5. Determine all values of $x$ in the interval $−2 < x < 6$ where $g$ has a relative maximum.

   Hint (e):

   Recall that a local maximum exists where the derivative changes from positive to negative. This can happen where $f^\prime(x) = 0$ or where $f^\prime(x) = \text{DNE}$.

   Hint (e):

   Notice that $g^\prime(x) = f(x)$; after all, the derivative of an integral is the original function.

6. Write the equation of the line tangent to $g$ at $x = 4$.

   Hint (f):

   What is the slope of $g(x)$ at $x = 4$ (see second hint in part (e))? What is the $y$-value?

7. Determine all values of $x$ in the interval $−2 < x < 6$ where $g$ has a point of inflection.

   Hint (g):

   Concavity is the slope of the slope. So inflection points are where the slope of the slope changes.