Question 1

Find the derivative of $f ( x ) = \sqrt { 4 - x }$ using the definition of the derivative.

Accepted Answer

The answer of Find the derivative of \(f ( x...

Question 2

Use the Intermediate Value Theorem to show that x3 - 5x - 7 = 0 for some value of x in (2, 3).

Accepted Answer

Let f (x) = x³ - 5x - 7. Sinc

Question 3

Suppose that f (x) is defined on [1, 3] and that f (1) = 3 and f (3) = 5. Sketch a possible graph of f that does not satisfy the conclusion of the Intermediate Value Theorem.

Accepted Answer

The answer of Suppose that f (x) is defined on...

Question 4

Given the graph of $y = f ^ { \prime } ( x )$ , answer the questions that follow.   (a) Find all values of x at which
(i) f is increasing.(iv) $f ^ { \prime \prime } ( x )  0$ (vi) $f ^ { \prime } \text { is increasing. }$ (ix) f has a local minimum.(b) Sketch a graph which could represent y = f (x).

Accepted Answer

(a) (i) f is increasing on [x₀, x₁]. (ii)

Question 5

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

Accepted Answer

A) -3 
B) -2 
C) -1 
D) 0 
E) 1
F)2
G)3
H)Does not exist 
A) -3 
B) -2 
C) -1 
D) 0 
E) 1
F)2
G)3
H)Does not exist

Question 6

Given the graph of $$y = f ^ { \prime } ( x )$$ below, select a graph which could be that of y = f (x).

Accepted Answer

A)    
B)    
C)    
D)    
E)   
F)    
G)   
H)
  
A)    
B)    
C)    
D)    
E)   
F)    
G)   
H)

Question 7

Let f (x) = $$\left\{ \begin{array} { l l } 
\sqrt { - 2 - x } & \text { if } x \leq - 2 \
x & \text { if } - 2  2
\end{array} \right.$$ . Find the following limits. Justify your answers.(a) $\lim _ { x \rightarrow - 2 ^ { - } } f ( x )$ (b) $\lim _ { x \rightarrow - 2 ^ { + } } f ( x )$ (c) $lim _ { x \rightarrow - 2 } f ( x )$ (d) $\lim _ { x \rightarrow 2 ^ { - } } f ( x )$ (e) $\lim _ { x \rightarrow 2 ^ { + } } f ( x )$ (f) $\lim _ { x \rightarrow 2 } f ( x )$

Accepted Answer

(a) @#IMG-DLM& (b) @#IMG-DLM&

Question 8

Find the value of the limit $\lim _ { x \rightarrow \infty } \frac { 7 + 3 x } { 4 - x }$

Accepted Answer

A) 
$\frac { 3 } { 4 }$ 
B) 3 
C) 7 
D)  $- \frac { 3 } { 4 }$ 
E) -3

F) $- \frac { 7 } { 4 }$
G)-7
H) $\frac { 7 } { 4 }$ 
A) 
$\frac { 3 } { 4 }$ 
B) 3 
C) 7 
D)  $- \frac { 3 } { 4 }$ 
E) -3

F) $- \frac { 7 } { 4 }$
G)-7
H) $\frac { 7 } { 4 }$

Question 9

Given the graph of f below, state the intervals on which f is continuous.

Accepted Answer

[-5, 1), (

Question 10

The following table shows the concentration (in mol/l) of a certain chemical in terms of reaction time (in hours) during a decomposition reaction.$$\begin{array} { | l | c | c | c | c | c | c | } 
\hline \text { Time (hours) } & 0 & 5 & 10 & 20 & 30 & 50 \
\hline \text { Concentration } ( \mathrm { mol } / \mathrm { L } ) & 2.32 & 1.86 & 1.49 & 0.98 & 0.62 & 0.25 \
\hline
\end{array}$$
(a) Find the average rate of change of concentration with respect to time for the following time intervals:
(i) [0; 5] (ii) [10; 20] (iii) [30; 50]
(b) Plot the points from the table and fit an appropriate exponential model to the data.(c) From your model in part (b), determine the instantaneous rate of change of concentration with respect to time.(d) Is the rate of change of concentration increasing or decreasing with respect to time? Justify your answer.

Accepted Answer

(a)
(i)
-0.092 mol/L/h
(ii)
-0

Question 11

Given the graph of y = f (x) below, select a graph which best represents the graph of $$y = f ^ { \prime } ( x )$$   
A)
  
B)  
C)   
D)  
E)
   
 F)
 
G)
  
H)

Accepted Answer

The answer of Given the graph of y = f...

Question 12

For the function f whose graph is given, arrange the following values in increasing order and explain your reasoning. $$f ^ { \prime } ( - 4 ) , f ^ { \prime } ( - 3 ) , f ^ { \prime } ( - 1 ) , f ^ { \prime } ( 0 ) , f ^ { \prime } ( 1 ) , f ^ { \prime } ( 2 ) , f ^ { \prime } ( 4 )$$

Accepted Answer

@#IMG-DLM& by inspec

Question 13

The graph of f is given below. State, with reasons, the number(s) at which
(a) f is not differentiable. (b) f is not continuous.

Accepted Answer

(a) f is not differentiable at x = -3 or

Question 14

$\text { Average Daily Temperatures in Moorhead, Minnesota }$
$\begin{array} { | l | c | c | c | c | c | c | } 
\hline \text { Month } & 1 ( \mathrm { Jan } ) & 2 ( \mathrm { Feb } ) & 3 ( \mathrm { Mar } ) & 4 ( \mathrm { Apr } ) & 5 ( \mathrm { May } ) & 6 ( \mathrm { Jun } ) \
\hline \text { Temperature } \left( { } ^ { \circ } \mathbf { F } \right) & 6 & 14 & 26 & 43 & 57 & 66 \
\hline \text { Month } & 7 ( \mathrm { Jul } ) & 8 ( \mathrm { Aug } ) & 9 ( \mathrm { Sep } ) & 10 ( \mathrm { Oct } ) & 11 ( \mathrm { Nov } ) & 12 ( \mathrm { Dec } ) \
\hline \text { Temperature } \left( { } ^ { \circ } \mathbf { F } \right) & 71 & 70 & 58 & 46 & 27 & 13 \
\hline
\end{array}$
-During which of the following periods was the rate of change of the average daily temperature the smallest?

Accepted Answer

A) [2, 6] 
B) [4, 8] 
C) [4, 9] 
D) [6, 10] 
E) [8, 10]
F)[9, 10]
G)[10, 11]
H)[11, 12] 
A) [2, 6] 
B) [4, 8] 
C) [4, 9] 
D) [6, 10] 
E) [8, 10]
F)[9, 10]
G)[10, 11]
H)[11, 12]

Question 15

A car on a test track accelerates from 0 ft/s to 208 ft/s in 8 seconds. The car’s velocity is given in the table below:
$\begin{array} { | c | c | c | c | c | c | c | c | c | c | } 
\hline t ( s ) & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \
\hline v ( t ) ( f t / s ) & 0 & 18 & 47 & 77 & 104 & 132 & 163 & 184 & 208 \
\hline
\end{array}$
-On what time interval does the car's average acceleration most closely approximate the average acceleration for the entire 8-second run?

Accepted Answer

A) [0, 1] 
B) [1, 2] 
C) [2, 3] 
D) [3, 4] 
E) [4, 5]
F)[5, 6]
G)[6, 7]
H)[7, 8] 
A) [0, 1] 
B) [1, 2] 
C) [2, 3] 
D) [3, 4] 
E) [4, 5]
F)[5, 6]
G)[6, 7]
H)[7, 8]

Question 16

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

Accepted Answer

The answer of Given the graph of y = f...

Question 17

Find the value of x at which the curve $y = \frac { x ^ { 2 } - 16 } { x ^ { 2 } - 5 x + 4 }$ has a vertical asymptote.

Accepted Answer

A) -4 
B) -3 
C) -2 
D) -1 
E) 0
F)1
G)2
H)3 
A) -4 
B) -3 
C) -2 
D) -1 
E) 0
F)1
G)2
H)3

Question 18

-For the function whose graph is given above, determine $\lim _ { x \rightarrow 3 } f ( x )$

Accepted Answer

A) -3 
B) -2 
C) -1 
D) 0 
E) 1
F)2
G)3
H)Does not exist 
A) -3 
B) -2 
C) -1 
D) 0 
E) 1
F)2
G)3
H)Does not exist

Question 19

Given the graph of y = f (x) below, select a graph which best represents the graph of $$y = f ^ { \prime } ( x )$$

Accepted Answer

A)    
B)    
C) 
  
D) 
  
E) 
  
F)   
G)   
H)
  
A)    
B)    
C) 
  
D) 
  
E) 
  
F)   
G)   
H)

Question 20

Explain why $\frac { ( 3 x - 2 ) ( x - 4 ) } { x - 4 } \neq 3 x - 2 , \text { but } \lim _ { x \rightarrow 4 } \frac { ( 3 x - 2 ) ( x - 4 ) } { x - 4 } = \lim _ { x \rightarrow 4 } ( 3 x - 2 )$

Accepted Answer

@#IMG-DLM& because the expression on the left does

Find the derivative of $f ( x ) = \sqrt { 4 - x }$ using the definition of the derivative.

Use the Intermediate Value Theorem to show that x³ - 5x - 7 = 0 for some value of x in (2, 3).

Suppose that f (x) is defined on [1, 3] and that f (1) = 3 and f (3) = 5. Sketch a possible graph of f that does not satisfy the conclusion of the Intermediate Value Theorem.

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

Given the graph of $y = f ^ { \prime } ( x )$ below, select a graph which could be that of y = f (x). $Given the graph of y = f ^ { \prime } ( x ) below, select a graph which could be that of y = f (x).$

Find the value of the limit $\lim _ { x \rightarrow \infty } \frac { 7 + 3 x } { 4 - x }$

Given the graph of f below, state the intervals on which f is continuous.

The graph of f is given below. State, with reasons, the number(s) at which (a) f is not differentiable. (b) f is not continuous.

$\text { Average Daily Temperatures in Moorhead, Minnesota }$ Month 1() 2() 3() 4() 5() 6() Temperature 6 14 26 43 57 66 Month 7() 8() 9() 10() 11() 12() Temperature 71 70 58 46 27 13 -During which of the following periods was the rate of change of the average daily temperature the smallest?

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

Find the value of x at which the curve $y = \frac { x ^ { 2 } - 16 } { x ^ { 2 } - 5 x + 4 }$ has a vertical asymptote.

$-For the function whose graph is given above, determine \lim _ { x \rightarrow 3 } f ( x )$ -For the function whose graph is given above, determine $\lim _ { x \rightarrow 3 } f ( x )$

Given the graph of y = f (x) below, select a graph which best represents the graph of $y = f ^ { \prime } ( x )$ $Given the graph of y = f (x) below, select a graph which best represents the graph of y = f ^ { \prime } ( x )$

Explain why $\frac { ( 3 x - 2 ) ( x - 4 ) } { x - 4 } \neq 3 x - 2 , \text { but } \lim _ { x \rightarrow 4 } \frac { ( 3 x - 2 ) ( x - 4 ) } { x - 4 } = \lim _ { x \rightarrow 4 } ( 3 x - 2 )$

Functions and Models

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Differential Equations

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

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Exam 2: Limits and Derivatives

Find the derivative of f(x)=4−xf ( x ) = \sqrt { 4 - x }f(x)=4−x​ using the definition of the derivative.

Use the Intermediate Value Theorem to show that x3 - 5x - 7 = 0 for some value of x in (2, 3).

Suppose that f (x) is defined on [1, 3] and that f (1) = 3 and f (3) = 5. Sketch a possible graph of f that does not satisfy the conclusion of the Intermediate Value Theorem.

-For the function whose graph is given above, determine lim⁡x→−2f(x)\lim _ { x \rightarrow - 2 } f ( x )limx→−2​f(x)

Given the graph of y=f′(x)y = f ^ { \prime } ( x )y=f′(x) below, select a graph which could be that of y = f (x).

Find the value of the limit lim⁡x→∞7+3x4−x\lim _ { x \rightarrow \infty } \frac { 7 + 3 x } { 4 - x }limx→∞​4−x7+3x​

Given the graph of f below, state the intervals on which f is continuous.

Given the graph of y = f (x) below, select a graph which best represents the graph of y=f′(x)y = f ^ { \prime } ( x )y=f′(x) A) B) C) D) E) F) G) H)

The graph of f is given below. State, with reasons, the number(s) at which (a) f is not differentiable. (b) f is not continuous.

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

Find the value of x at which the curve y=x2−16x2−5x+4y = \frac { x ^ { 2 } - 16 } { x ^ { 2 } - 5 x + 4 }y=x2−5x+4x2−16​ has a vertical asymptote.

-For the function whose graph is given above, determine lim⁡x→3f(x)\lim _ { x \rightarrow 3 } f ( x )limx→3​f(x)

Given the graph of y = f (x) below, select a graph which best represents the graph of y=f′(x)y = f ^ { \prime } ( x )y=f′(x)

Functions and Models

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Differential Equations

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Filters

Find the derivative of $f ( x ) = \sqrt { 4 - x }$ using the definition of the derivative.

Use the Intermediate Value Theorem to show that x³ - 5x - 7 = 0 for some value of x in (2, 3).

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

Given the graph of $y = f ^ { \prime } ( x )$ below, select a graph which could be that of y = f (x). $Given the graph of y = f ^ { \prime } ( x ) below, select a graph which could be that of y = f (x).$

Find the value of the limit $\lim _ { x \rightarrow \infty } \frac { 7 + 3 x } { 4 - x }$

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

Find the value of x at which the curve $y = \frac { x ^ { 2 } - 16 } { x ^ { 2 } - 5 x + 4 }$ has a vertical asymptote.

$-For the function whose graph is given above, determine \lim _ { x \rightarrow 3 } f ( x )$ -For the function whose graph is given above, determine $\lim _ { x \rightarrow 3 } f ( x )$

Given the graph of y = f (x) below, select a graph which best represents the graph of $y = f ^ { \prime } ( x )$ $Given the graph of y = f (x) below, select a graph which best represents the graph of y = f ^ { \prime } ( x )$