Exam 10: Introduction to the Derivative

The cost of fighting crime in the U.S. increased steadily in the period 1982 - 1999. Total spending on police, courts, and prisons can be approximated, respectively, by $P ( t ) = 1.745 t + 29.84$ billion dollars $( 2 \leq t \leq 19 )$ $C ( t ) = 1.097 t + 10.65$ billion dollars $( 2 \leq t \leq 19 )$ $J ( t ) = 1.919 t + 12.36$ billion dollars $( 2 \leq t \leq 19 )$ Where t is time in years since 1980. Compute $\lim _ { t \rightarrow \infty } \frac { P ( t ) } { P ( t ) + C ( t ) + J ( t ) }$ to two decimal places.

Calculate the average rate of change of the given function (Inflation (%) of Budget deficit (% of GNP)) over the interval $[ 0,2 ]$ . ​ ​ ​ Please enter your answer as a number without the units.

Compute $f ^ { \prime } ( a )$ for $a = 6$ . $f ( x ) = \frac { 5 } { x }$

Estimate the limit numerically. $\lim _ { x \rightarrow + \infty } \frac { 6 x ^ { 4 } + 17 x ^ { 3 } } { 1,447 x ^ { 6 } + 7 }$

Use a graph of f or some other method to determine what, if any, value to assign to $f ( a )$ to make f continuous at $x = a$ . $f ( x ) = \frac { x ^ { 2 } - 2 x } { x + 5 }$ ; $a = - 5$

The function given below gives the cost to manufacture x items. Estimate (using $h = 0.0001$ ) the instantaneous rate of change of the total cost at the production level $x = 80$ . ​ $C ( x ) = 14,700 + 100 x + \frac { 900 } { x }$ ​ Enter your answer as a number without the units rounded to the nearest tenth.

Calculate the limit algebraically. $\lim _ { x \rightarrow + \infty } \frac { 7 x ^ { 2 } + 7 x - 6 } { 10 x ^ { 2 } - 2 }$

The graph of a function f is given. Determine whether f is continuous on its domain. ​ ​

Determine what, if any, value to assign to $f ( a )$ to make f continuous at $x = a$ . $f ( x ) = \frac { 1 - e ^ { x } } { x }$ ; $a = 0$

Calculate the average rate of change of the given function f over the interval $[ a , a + h ]$ , where $h = 1,0.1,0.01,0.001$ , and $0.0001$ . (It will be easier to do this if you first simplify the difference quotient (dq) as much as possible.) $f ( x ) = 5 x ^ { 2 } ; a = 0$ Complete the table. dq 1 0.1 0.01 0.001 0.0001

Estimate the derivative of the function $f ( x ) = \frac { x } { 3 } - 3$ at the point $x = 3$ .

Find all points of discontinuity of the function. $g ( x ) = \left\{ \begin{aligned}x + 5 & \text { if } x < 0 \\3 x + 5 & \text { if } 0 \leq x < 4 \\x ^ { 2 } + 5 & \text { if } x \geq 4\end{aligned} \right.$

The function $R ( t ) = 42 t - t ^ { 2 }$ represents the value of the U.S. dollar in Indian rupees as a function of time t in days.Find the average rates of change of $R ( t )$ over the time intervals $[ t , t + h ]$ , where t is as indicated and $h = 1$ , $0.1$ , $0.01$ , and $0.0001$ days. Hence, estimate (using $h = 0.0001$ ) the instantaneous rate of change of R at time $t = 7$ . Please round the instantaneous rate to the nearest whole number.

Use a graph to determine whether the given function is continuous on its domain. If it is not continuous on its domain, list the points of discontinuity. $g ( x ) = \frac { x - 4 } { x + 2 }$

Calculate the limit algebraically. ​ $\lim _ { x \rightarrow - \infty } \frac { 11 x ^ { 5 } - 7,000 x ^ { 4 } } { 8 x ^ { 5 } + 70,000 }$

If a stone is dropped from a height of Variable $958$ isn't defined feet, its height after t seconds is given by $s = 958 - 22 t ^ { 2 }$ . Find the stone's velocity at time $t = 4$ .

Find an equation of the tangent line to the graph of the function $f ( x ) = 2 \sqrt { x }$ at the point that has x-coordinate $x = 36$ . [Hint: use point-slope formula to find the equation of the tangent line.]

Determine what, if any, value to assign to $f ( a )$ to make f continuous at $x = a$ . $f ( x ) = \frac { x ^ { 2 } - 6 x + 8 } { x - 4 }$ ; $a = 4$

Calculate the average rate of change of the given function (Inflation (%) of Budget deficit (% of GNP)) over the interval $[ 0,2 ]$ .

Find all points of discontinuity of the function. $h ( x ) = \left\{ \begin{aligned}4 - x & \text { if } x < 4 \\3 & \text { if } x = 4 \\x + 1 & \text { if } x > 4\end{aligned} \right.$