Exam 8: Limits and Derivatives

You deposit $\$$ 5000 in an account with an annual interest rate of change r (in decimal form) compounded monthly. At the end of 5 years, the balance is $A = 5000 \left( 1 + \frac { r } { 12 } \right) ^ { 60 }$ . Find the rates of change of A with respect to r when $r = 0.08$ .

The profit (in dollars) from selling x units of calculus textbooks is given by $p = - 0.05 x ^ { 2 } + 20 x - 1000$ . Find the marginal profit when $x = 149$ . Round your answer to two decimal places.

Determine whether the given function is continuous. If it is not, identify where it is discontinuous. $y = 7 x ^ { 2 } - 9 x + 3$

Find the derivative of the function. $f ( x ) = x ^ { 5 }$

Find the derivative of the function $f ( x ) = \frac { x ^ { 2 } - x - 20 } { x + 4 }$ . State which differentiation rule(s) you used to find the derivative.

Sketch the graph of the function $f ( x ) = \frac { x ^ { 2 } - 4 } { x - 2 }$ and describe the interval(s) on which the function is continuous.

Complete the table and use the result to estimate the limit. $\lim _ { x \rightarrow - 2 } \frac { \frac { 1 } { x - 10 } + \frac { 1 } { 12 } } { x + 2 }$ x -2.1 -2.01 -2.001 -1.999 -1.99 -1.9 f(x)

The graph shows the number of visitors V to a national park in hundreds of thousands during a one-year period, where t = 1 represents January. Estimate the rate of change of V over the interval $[ 9,12 ]$ . Round your answer to the nearest hundred thousand visitors per year.

Find the slope of the tangent line to the graph of the function at the given point. $f ( x ) = 4 x ^ { 2 } + 4 , \quad ( 3,40 )$

Find the x-values (if any) at which the function $f ( x ) = \frac { x } { x ^ { 2 } + 16 }$ is not continuous. Which of the discontinuities are removable?

The profit (in dollars) from selling x units of calculus textbooks is given by $p = - 0.05 x ^ { 2 } + 20 x - 3000$ . Find the additional profit when the sales increase from 149 to 150 units. Round your answer to two decimal places.

Find the limit (if it exists): $\lim _ { \Delta x \rightarrow 0 } \frac { ( x + \Delta x ) ^ { 2 } - 13 ( x + \Delta x ) + 13 - \left( x ^ { 2 } - 13 x + 13 \right) } { \Delta x }$

Differentiate the given function. $y = \frac { 7 } { ( 6 x ) ^ { 6 } }$

The value V of a machine $t$ years after it is purchased is inversely proportional to the square root of $t + 5$ . The initial value of the machine is $\$$ 10,000. Find the rate of depreciation when $t = 3$ . Round your answer to two decimal places.

The cost (in dollars) of removing $p\%$ of the pollutants from the water in a small lake is given by $C = \frac { 23,000 p } { 300 - p } , 0 \leq p < 300$ . Evaluate $\lim _ { p \rightarrow 300 ^ { - } } C$ .

Suppose that $\lim _ { x \rightarrow c } f ( x ) = 11$ and $\lim _ { x \rightarrow c } g ( x ) = - 15$ . Find the following limit: $\lim _ { x \rightarrow c } [ f ( x ) + g ( x ) ]$

Use the limit definition to find the slope of the tangent line to the graph of $f ( x ) = \sqrt { 4 x + 25 }$ at the point $( 6,7 )$ .

Describe the interval $($ s $)$ on which the function $f ( x ) = \frac { x - 4 } { x ^ { 2 } - 16 }$ is continuous.

Find the marginal cost for producing x units. (The cost is measured in dollars.) $C = 205,000 + 9800 x$

Find the derivative of the function. $f ( x ) = x ^ { 4 } ( 3 + 3 x ) ^ { 3 }$