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

Match the function with the correct table values. - $f ( x ) = \frac { x ^ { 2 } - 9 } { x - 3 }$

Solve the problem. -The position of an object at time $t$ is given by $s ( t )$ . Find the instantaneous velocity at the indicated value of $t$ . $s ( t ) = \frac { 3 } { t - 1 } \quad$ at $t = 4$

Use the given graph to determine the limit, if it exists. -Let $f ( x ) = \left\{ \begin{array} { c } - 4 , x > 3 \\ - \frac { x } { 4 } , x < 3 \\ 4 , x = 3 \end{array} \right.$ $\lim _ { x \rightarrow } f ( x )$

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. - $\int _ { 0 } ^ { 6 \pi } \sin \left( \frac { x } { 6 } \right) d x$ (Hint: each rectangle is 6 times as wide.)

Use the given graph to determine the limit, if it exists. - Find $\lim _ { x \rightarrow 1 ^ { - } } f ( x )$ and $\lim _ { x \rightarrow 1 ^ { + } } f ( x )$ .

Use NINT on a calculator to find the numerical integral of the function over the specified interval. - $f ( x ) = \frac { 1 } { x } ; \left[ e ^ { 4 } , e ^ { 7 } \right]$

Estimate the slope of the tangent line at the indicated point. -

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. - $\int _ { 0 } ^ { \pi } ( \sin x + 1 ) d x$

Compute the average of the RRAM and LRAM approximations to estimate the area between the graph of the function and the x-axis over the given interval using the indicated number of subintervals. (The function is non-negative on the given interval). - $f ( x ) = 3 x + 3 ; [ 0,5 ] ; 5$ subintervals

Solve the problem. -A rocket is launched straight up from level ground. The distance (in $\mathrm { ft }$ ) of the rocket above the ground (the position function) is $\mathrm { f } ( \mathrm { t } ) = 224 \mathrm { t } - 16 \mathrm { t } ^ { 2 }$ at any time $\mathrm { t }$ (in sec). Find $\mathrm { f } ^ { \prime } ( 3 )$ and the initial velocity of the rocket.

Use the given graph to determine the limit, if it exists. -Let $f ( x ) = \left\{ \begin{array} { l l } 3 x + 4 , & x < - 4 \\ x - 4 , & x > - 4 \end{array} \right.$ $\lim _ { x \rightarrow 4 } f ( x )$

Use the given graph to determine the limit, if it exists. - $\lim _{x \rightarrow-1} f(x)$

Find the limit of the function by using direct substitution. - $\lim _ { x \rightarrow a } \frac { x ^ { 2 } - 2 x - 15 } { x + 3 }$

Match the function with the correct table values. - $f ( x ) = \sqrt { x } - 2$

Solve the problem. -The velocity of a projectile fired straight into the air is given every half second. Use left endpoints to estimate the distance the projectile traveled in four seconds. TimdVelocity

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. - $\int _ { 0 } ^ { \pi } 7 \sin x d x$

Solve the problem. -Estimate the "RRAM" area under the graph of the function above the $x$ -axis and under the graph of the function from $x = 0$ to $x = 5$ . Use 5 subintervals.

Sketch a possible graph for a function f that has the stated properties. - $\text { The domain of } \mathrm { f } \text { is } [ - 5,0 ] \text { and the derivative at } \mathrm { x } = - 2 \text { is } 3 \text {. }$

Use NDER on a calculator to find the numerical derivative of the function at the specified point. - $f ( x ) = 5 x ^ { 2 } + x \text { at } x = - 4$

Use the given graph to determine the limit, if it exists. - Find $\lim _ { x - \underline { 2 } ^ { - } } f ( x )$ and $\lim _ { x - \underline { - } ^ { + } } f ( x )$ .