Exam 2: Limits and Derivatives

If f (x) = 5, which of the following represents $f ^ { \prime } ( 3 ) ?$ A) $\lim _ { \Delta x \rightarrow 0 } \frac { 5 } { \Delta x }$ B)3 C) $\lim _ { \Delta x \rightarrow 0 } \frac { 5 - 3 } { \Delta x }$ E) $\lim _ { \Delta x \rightarrow 0 } \frac { 5 - 5 } { \Delta x }$ D)5 F) $\lim _ { \Delta x \rightarrow 0 } \frac { 3 } { \Delta x }$ G)Does not exist H)None of the above

Given the following information about limits, sketch a graph which could be the graph of y = f (x). Label all horizontal and vertical asymptotes. f(x)=f(x)=-1,f(x)=f(x)=-\infty f(x)=f(x)=\infty, and f=(0)=1

Given that $\lim _ { x \rightarrow 3 } f ( x ) = 5 , \lim _ { x \rightarrow 3 } g ( x ) = 0 , \text { and } \lim _ { x \rightarrow 3 } h ( x ) = - 8 \text {, }$ find the following limits, if they exist. If a limit does not exist, explain why.(a) $\lim _ { x \rightarrow 3 } ( f ( x ) + h ( x ) )$ (b) $\lim _ { x \rightarrow 3 } x ^ { 2 } f ( x )$ (c) $\lim _ { x \rightarrow 3 } f ^ { 2 } ( x )$ (d) $\lim _ { x \rightarrow 3 } \frac { f ( x ) } { 2 h ( x ) }$ (e) $\lim _ { x \rightarrow 3 } \frac { g ( x ) } { f ( x ) }$ (f) $\lim _ { x \rightarrow 3 } \frac { f ( x ) } { g ( x ) }$ (g) $\lim _ { x \rightarrow 3 } \frac { 2 h ( x ) } { f ( x ) - h ( x ) }$ (h) $\lim _ { x \rightarrow 3 } \sqrt [ 3 ] { h ( x ) }$

Find an equation of the line tangent to the curve y = x + (1/x) at the point $\left( 5 , \frac { 26 } { 25 } \right)$

-For the function whose graph is given above, determine $\lim _ { x \rightarrow - 2 } f ( x )$

Find the derivative of f (x) = 3x - 5 using the definition of the derivative.

Use the given graph to find the indicated quantities: (a) $\lim _ { x \rightarrow - 1 ^ { - } } f ( x )$ (b) $\lim _ { x \rightarrow - 1 ^ { + } } f ( x )$ (c) $\varliminf _ { x \rightarrow - 1 } f ( x )$ (d) $\lim _ { x \rightarrow 1 ^ { - } } f ( x )$ (e) $\lim _ { x \rightarrow 1 ^ { + } } f ( x )$ (f) $\lim _ { x \rightarrow 1 } f ( x )$ (g) $\lim _ { x \rightarrow 2 ^ { - } } f ( x )$ (h) $\lim _ { x \rightarrow 2 ^ { + } } f ( x )$ (i) $\lim _ { x \rightarrow 2 } f ( x )$ (j) f (-1) (k) f (0) (l) f (1) (m) f (2)

Find the distance between the two values of x at which the function $\frac { 1 } { x ^ { 2 } - 3 x + 2 }$ is discontinuous.

Given the graph of y = f (x), sketch the graph of y = $f ^ { \prime } ( x )$ (a) (b) (c) (d)

Find the value of the limit $\lim _ { x \rightarrow - 2 } \left( \frac { 1 } { x + 2 } + \frac { 4 } { x ^ { 2 } - 4 } \right)$

Below are the graphs of a function and its first and second derivatives. Identify which of the following graphs (a, b, and c) is f (x), which is $f ^ { \prime } ( x )$ , and which is $f ^ { \prime \prime } ( x )$ . Justify your choices.

In order to determine an appropriate delivery schedule to a group of rural homes in North Dakota, a fuel oil distributor monitors fuel oil consumption and the daily outdoor temperature (in degrees Fahrenheit). A table was constructed for a function F (T) of fuel oil consumption (in gallons per day) as a function of the temperature T.(a) Sketch a graph which you believe would approximate the graph of y = F (T).(b) What is the meaning of $F ^ { \prime } ( T ) ?$ What are its units? (c) Write a sentence that would explain to an intelligent layperson the meaning of $F ^ { \prime } ( 0 ) = - 0.4$

The displacement in meters of a particle moving in a straight line is given by s = t³, where t is measured in seconds. Find the average velocity in meters per second over the time period [1, 2].

For each of the following problems, make an appropriate table to determine the limits.(a) $\lim _ { x \rightarrow 1 } \frac { | x | - x } { x - 1 }$ (b) $\lim _ { x \rightarrow 0 } \frac { \cos x - 1 } { x }$ (c) $\lim _ { x \rightarrow 3 } \frac { ( 1 / x ) - ( 1 / 3 ) } { x - 3 }$

For the curve f(x) = $\sqrt { x + 3 }$ , find the slope M_PQ of the secant line through the points P = (1, f (1)) and Q = (6, f (6)).

Given the graph of y = f (x) below, sketch the graph of $y = f ^ { \prime } ( x )$ .

Suppose that the height of a projectile red vertically upward from a height of 80 feet with an initial velocity of 64 feet per second is given by h (t) = -16t² + 64t + 80.(a) Compute the height of the object for t = 0, 1, 2, 3, 4, 5, and 6 seconds.(b) What is the physical significance of h (6)? What does that suggest about the domain of h? (c) What is the average velocity of the projectile for each of the following time intervals? (i) [1; 3] (ii) [0; 2] (iii) [0; 4] (d) What is the physical significance of an average velocity of 0? (e) When does the projectile reach its maximum height? (f) For what value(s) of t is h (t) = 0? Are all solutions to the equation valid? Explain.

A weight is attached to a spring. Suppose the position (in meters) of the weight above the floor t seconds after it is released is given by $P ( t ) = 0.5 \sin \left( \pi t + \frac { \pi } { 2 } \right) + 1.2$ . What is the average rate of change of the position of the weight (in m/s) over the time interval [3, 5]?

If f (-1) = 3 and $f ^ { \prime } ( - 1 ) = 2$ , find an equation of the tangent line at x = -1.

Given the graph of y = f (x), find all values of x for which (a) $f ^ { \prime } ( x ) > 0$ (b) $f ^ { \prime } ( x ) < 0$ (c) $f ^ { \prime } ( x ) = 0$ (d) $f ^ { \prime \prime } ( x ) > 0$