Exam 11: An Introduction to Calculus: Limits, Derivatives, and Integrals

Solve the problem. -The table shows the velocity of a remote controlled race car moving along a dirt path for 8 seconds. Estimate the distance traveled by the car using 8 subintervals of length 1 with right-end point values.

It can be shown that the area enclosed between the x-axis and one arch of the sine curve is 2. Use this fact to compute the definite integral. - $\int _ { - \pi / 2 } ^ { \pi / 2 } ( \cos x + 3 ) d x$

Find the definite integral by computing an area. - $\int _ { 0 } ^ { 8 } ( k x + 3 ) d x ; k \geq 0$

Find the limit. -Let $\lim _ { x \rightarrow 6 } f ( x ) = - 5$ and $\lim _ { x \rightarrow 6 } g ( x ) = 4$ . Find $\lim _ { x \rightarrow 6 } \frac { - 4 f ( x ) - 6 g ( x ) } { - 9 + g ( x ) }$ .

Sketch a possible graph for a function f that has the stated properties. - $\text { The domain of } \mathrm { f } \text { is } [ 0,5 ] \text { and the derivative at } x = 1 \text { is } 1 \text {. }$

Find the derivative of the function at the specified point. - $f ( x ) = 3 x ^ { 2 } + 10 x$ at $x = 8$

Find the derivative of the function using the definition of derivative. - $f ( x ) = x ^ { 3 } + 5 x$