Exam 10: Introduction to the Derivative

Calculate the average rate of change of the given function (Inflation(%)) over the interval $[ 1,4 ]$ . ​ ​ ​ Please enter your answer as a fraction without the units.

Compute $f ^ { \prime } ( a )$ algebraically for $a = - 1$ . ​ $f ( x ) = 10 x + 5$

Use the graph to compute $\lim _ { x \rightarrow 3 } f ( x )$ and $f ( 3 )$ .

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 $x = a$ . ​ $f ( x ) = \frac { x - 4 } { x ^ { 3 } - 64 }$ ; $a = 4$

Calculate the average rate of change of the given function over the interval $[ - 5 , - 3 ]$ . -5 -4 -3 -2 f(x) -10.9 -0.5 -1.1 -7.2

Estimate $g ^ { \prime } ( 3 )$ of the function $g ( t ) = \frac { 1 } { t ^ { 2 } }$ Please enter your answer as a fraction.

Use the graph to compute the given quantity. $\lim _ { x \rightarrow 2 } f ( x )$

Estimate the limit numerically. $\lim _ { x \rightarrow 6 } \frac { x ^ { 2 } - 1 } { x - 6 }$

Estimate $\left. \frac { \mathrm { d } y } { \mathrm {~d} x } \right| _ { x = - 1 }$ of the function $y = 3 - x ^ { 2 }$ .

Calculate the average rate of change of the given function over the interval $[ 1,3 ]$ . 0 1 2 3 f(x) 4 15 5 -9

Compute the derivative function $f ^ { \prime } ( x )$ algebraically. ​ $f ( x ) = x ^ { 2 } - 3$

On January 1, 1996, America Online was the largest on-line service provider, with 5 million subscribers, and it was adding new subscribers at a rate of 42,000 per week. If $A ( t )$ is the number of America Online subscribers t weeks after January 1, 1996, what do the given data tell you about values of the function $A ( t )$ and its derivative

Compute the derivative function $f ^ { \prime } ( x )$ algebraically. $f ( x ) = - 5 x + 7$

Compute the derivative function $f ^ { \prime } ( x )$ algebraically. $f ( x ) = \frac { 2 } { x }$

Calculate the average rate of change of the given function over the interval $[ 4,4.5 ]$ . p() 4 4.5 5 q(p) (items) 500 100 150

Estimate $\left. \frac { \mathrm { d } R } { \mathrm {~d} p } \right| _ { p = 17 }$ of the function $R = \frac { 16 } { p }$ .

Calculate the average rate of change of the given function over the interval $[ 7,8 ]$ . $f ( x ) = \frac { 168 } { x }$

The function $R ( t ) = 140 + 3 t ^ { 3 }$ 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. Estimate (using $h = 0.0001$ ) the instantaneous rate of change of R at time $t = 7$ . Please select the correct answer rounded to the nearest whole number.

Calculate the limit algebraically. $\lim _ { h \rightarrow 0 } \frac { 12 h ^ { 2 } } { h + 13 h ^ { 2 } }$