Question 1

Use differentiation to determine whether the integral formula is correct.
-$$\int ( 2 x - 2 ) ^ { 4 } d x = \frac { ( 2 x - 2 ) ^ { 5 } } { 10 } + C$$

Accepted Answer

A) True 
 B)False

Question 2

Find an antiderivative of the given function.
-$\cos \pi x + 6 \sin \frac { x } { 6 }$

Accepted Answer

A)  $\frac { 1 } { \pi } \sin \pi x - \cos \frac { x } { 6 }$ 
B)  $- \pi \sin \pi x + \cos \frac { x } { 6 }$ 
C)  $- \sin \pi x - 36 \cos \frac { x } { 6 }$ 
D)  $\frac { 1 } { \pi } \sin \pi x - 36 \cos \frac { x } { 6 }$ 
A)  $\frac { 1 } { \pi } \sin \pi x - \cos \frac { x } { 6 }$ 
B)  $- \pi \sin \pi x + \cos \frac { x } { 6 }$ 
C)  $- \sin \pi x - 36 \cos \frac { x } { 6 }$ 
D)  $\frac { 1 } { \pi } \sin \pi x - 36 \cos \frac { x } { 6 }$

Question 3

Solve the problem.
-The graphs below show the first and second derivatives of a function $y = f ( x )$. Select a possible graph $f$ that passe through the point $P$.

Accepted Answer

A) 
 
[NOTE: Graph vertical scales
may vary from graph to graph.] 
B) 
 
[NOTE: Graph vertical scales
may vary from graph to graph.] 
C) 
 
[NOTE: Graph vertical scales
may vary from graph to graph.] 
D)

[NOTE: Graph vertical scales
may vary from graph to graph.] 
A) 
 
[NOTE: Graph vertical scales
may vary from graph to graph.] 
B) 
 
[NOTE: Graph vertical scales
may vary from graph to graph.] 
C) 
 
[NOTE: Graph vertical scales
may vary from graph to graph.] 
D)

[NOTE: Graph vertical scales
may vary from graph to graph.]

Question 4

Solve the problem.
-Use Newton's method to find the value of $x$ for which $\ln ( 1 / x ) = 1 - x ^ { 2 }$. Find the answer correct to five decimal places.

Accepted Answer

A)  0.45076 
B)  0.45071 
C)  0.45065 
D)  0.45081 
A)  0.45076 
B)  0.45071 
C)  0.45065 
D)  0.45081

Question 5

Use l'H^opital's rule to find the limit.
-$\lim _ {\theta  \rightarrow \ - 0 } \frac { 3 - 3 \cos \theta } { \sin 4 \theta }$

Accepted Answer

A)  $\infty$ 
B)  0 
C)  1 
D)  $\frac { 3 } { 4 }$ 
A)  $\infty$ 
B)  0 
C)  1 
D)  $\frac { 3 } { 4 }$

Question 6

Find the location of the indicated absolute extremum for the function.
-

Accepted Answer

A)  No maximum 
B)  x = 0 
C)  x = -1 
D)  x = 2 
A)  No maximum 
B)  x = 0 
C)  x = -1 
D)  x = 2

Question 7

Find the location of the indicated absolute extremum for the function.
-

Accepted Answer

A)  No minimum 
B)  x = -1 
C)  x = 2 
D)  x = 1 
A)  No minimum 
B)  x = -1 
C)  x = 2 
D)  x = 1

Question 8

Answer the question.
-A marathoner ran the 26.2 mile New York City Marathon in 2.8 hrs. Did the runner ever exceed a speed of 9
miles per hour?

Accepted Answer

Yes, the Mean Value Theorem im

Question 9

Solve the problem.
-$$\text { Show that } g ( x ) = \frac { a + x } { \sqrt { b ^ { 2 } + ( a + x ) ^ { 2 } } } \text { is an increasing function of } x$$

Accepted Answer

The answer of Solve the problem.
-\[\text { Show that }...

Question 10

Graph the rational function.
-\9\begin{array} { c } 
y = \frac { x ^ { 2 } + x - 42 } { x ^ { 2 } - x - 56 } \

\end{array}\)

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 11

Find an antiderivative of the given function.
-$$x ^ { - 5 } + \frac { 1 } { 4 \sqrt { x } }$$

Accepted Answer

A)  $- \frac { 1 } { 5 x ^ { 5 } } + \frac { 1 } { 2 } x ^ { 1 / 2 }$ 
B)  $- \frac { 1 } { 4 x ^ { 5 } } + \frac { 1 } { 2 } x ^ { 1 / 2 }$ 
C)  $- \frac { 1 } { 4 x ^ { 4 } } + \frac { 1 } { 2 } x ^ { 1 / 2 }$ 
D)  $- \frac { 1 } { 5 x ^ { 4 } } + \frac { 1 } { 2 } x ^ { 1 / 2 }$ 
A)  $- \frac { 1 } { 5 x ^ { 5 } } + \frac { 1 } { 2 } x ^ { 1 / 2 }$ 
B)  $- \frac { 1 } { 4 x ^ { 5 } } + \frac { 1 } { 2 } x ^ { 1 / 2 }$ 
C)  $- \frac { 1 } { 4 x ^ { 4 } } + \frac { 1 } { 2 } x ^ { 1 / 2 }$ 
D)  $- \frac { 1 } { 5 x ^ { 4 } } + \frac { 1 } { 2 } x ^ { 1 / 2 }$

Question 12

Solve the problem.
-At about what velocity do you enter the water if you jump from a 15 meter cliff? (Use $g = 9.8 \mathrm {~m} / \mathrm { sec } ^ { 2 }$.)

Accepted Answer

A)  $- 17 \mathrm {~m} / \mathrm { sec }$ 
B)  $- 8.5 \mathrm {~m} / \mathrm { sec }$ 
C)  $17 \mathrm {~m} / \mathrm { sec }$ 
D)  $2 \mathrm {~m} / \mathrm { sec }$ 
A)  $- 17 \mathrm {~m} / \mathrm { sec }$ 
B)  $- 8.5 \mathrm {~m} / \mathrm { sec }$ 
C)  $17 \mathrm {~m} / \mathrm { sec }$ 
D)  $2 \mathrm {~m} / \mathrm { sec }$

Question 13

Show that the function has exactly one zero in the given interval.
-$f ( x ) = x ^ { 3 } + \frac { 4 } { x ^ { 2 } } + 1 , ( - \infty , 0 )$

Accepted Answer

The function @#LAT-DLM& is continuous on the open

Question 14

Find a value of c that makes the function continuous at the given value of x. If it is impossible, state this.
-$$f ( x ) = \left\{ \begin{array} { c c } 
( \sin x ) ^ { x } , & x \neq 0 \
c , & x = 0
\end{array} ; \quad x = 0 \right.$$

Accepted Answer

A)  -1 
B)  0 
C)  1 
D)  Impossible 
A)  -1 
B)  0 
C)  1 
D)  Impossible

Question 15

Answer the problem.
-Let $f ( x ) = ( x - 1 ) ^ { 2 / 3 }$
(a) Does $f ^ { \prime } ( 1 )$ exist?
(b) Show that the only local extreme value of $f$ occurs at $x = 1$.
(c) Does the result of (b) contradict the Extreme Value Theorem?
(d) Repeat parts (a) and (b) for $f ( x ) = ( x - c ) ^ { 2 / 3 }$.
Give reasons for your answers.

Accepted Answer

(a) No, since @#LAT-DLM&, which is undefined at @#LAT-DLM&.

Question 16

Solve the problem.
-Suppose Newton's Method is used with an initial guess $x _ { 0 }$ that lies at a critical point $( a , b ) , b \neq 0$. What happens to $x _ { 1 }$ and later approximations? Give reasons for your answer.

Accepted Answer

The answer of Solve the problem.
-Suppose Newton's Method is used...

Question 17

Solve the problem.
-Write down the first four approximations to the solution of the equation $\sin 3 x = x$ using Newton's method with an initial estimate of $x _ { 0 } = 1$.

Accepted Answer

\[\begin{array} { l } 
x _ { 0

Question 18

Find a value of c that makes the function continuous at the given value of x. If it is impossible, state this.
-$$f ( x ) = \left\{ \begin{array} { c l } 
\frac { 4 x - 2 \sin 2 x } { 3 x ^ { 3 } } , & x \neq 0 \
c , & x = 0
\end{array} ; x = 0 \right.$$

Accepted Answer

A)  4 
B)  0 
C)  $\frac { 4 } { 9 }$ 
D)  $\frac { 8 } { 9 }$ 
A)  4 
B)  0 
C)  $\frac { 4 } { 9 }$ 
D)  $\frac { 8 } { 9 }$

Question 19

Solve the initial value problem.
-$$\frac { d y } { d x } = \frac { 1 } { x + 5 } , y ( - 4 ) = 6$$

Accepted Answer

A)  $y = \ln ( x + 5 ) + 6$ 
B)  $y = - \frac { 1 } { ( x + 5 ) ^ { 2 } } + 6$ 
C)  $y = - \frac { 1 } { ( x + 5 ) ^ { 2 } } + 7$ 
D)  $y = \ln ( x + 5 ) + 6 - \ln 2$ 
A)  $y = \ln ( x + 5 ) + 6$ 
B)  $y = - \frac { 1 } { ( x + 5 ) ^ { 2 } } + 6$ 
C)  $y = - \frac { 1 } { ( x + 5 ) ^ { 2 } } + 7$ 
D)  $y = \ln ( x + 5 ) + 6 - \ln 2$

Question 20

Solve the problem.
-Given the velocity and initial position of a body moving along a coordinate line at time $t$, find the body's position $\mathrm { t }$ $v = - 16 t + 4 , s ( 0 ) = 7$

Accepted Answer

A)  $s = - 8 t ^ { 2 } + 4 t - 7$ 
B)  $s = - 8 t ^ { 2 } + 4 t + 7$ 
C)  $s = - 16 t ^ { 2 } + 4 t + 7$ 
D)  $s = 8 t ^ { 2 } + 4 t - 7$ time 
A)  $s = - 8 t ^ { 2 } + 4 t - 7$ 
B)  $s = - 8 t ^ { 2 } + 4 t + 7$ 
C)  $s = - 16 t ^ { 2 } + 4 t + 7$ 
D)  $s = 8 t ^ { 2 } + 4 t - 7$ time

Use differentiation to determine whether the integral formula is correct. - $\int ( 2 x - 2 ) ^ { 4 } d x = \frac { ( 2 x - 2 ) ^ { 5 } } { 10 } + C$

Find an antiderivative of the given function. - $\cos \pi x + 6 \sin \frac { x } { 6 }$

Solve the problem. -The graphs below show the first and second derivatives of a function $y = f ( x )$ . Select a possible graph $f$ that passe through the point $P$ .

Solve the problem. -Use Newton's method to find the value of $x$ for which $\ln ( 1 / x ) = 1 - x ^ { 2 }$ . Find the answer correct to five decimal places.

Use l'H^opital's rule to find the limit. - $\lim _ {\theta \rightarrow \ - 0 } \frac { 3 - 3 \cos \theta } { \sin 4 \theta }$

Find the location of the indicated absolute extremum for the function. -

Find the location of the indicated absolute extremum for the function. -

Answer the question. -A marathoner ran the 26.2 mile New York City Marathon in 2.8 hrs. Did the runner ever exceed a speed of 9 miles per hour?

Solve the problem. - $\text { Show that } g ( x ) = \frac { a + x } { \sqrt { b ^ { 2 } + ( a + x ) ^ { 2 } } } \text { is an increasing function of } x$

Graph the rational function. -\9\begin{array} { c } y = \frac { x ^ { 2 } + x - 42 } { x ^ { 2 } - x - 56 } \\ \end{array}\) $Graph the rational function. -\9\begin{array} { c } y = \frac { x ^ { 2 } + x - 42 } { x ^ { 2 } - x - 56 } \\ \end{array}\)$

Find an antiderivative of the given function. - $x ^ { - 5 } + \frac { 1 } { 4 \sqrt { x } }$

Solve the problem. -At about what velocity do you enter the water if you jump from a 15 meter cliff? (Use $g = 9.8 \mathrm {~m} / \mathrm { sec } ^ { 2 }$ .)

Show that the function has exactly one zero in the given interval. - $f ( x ) = x ^ { 3 } + \frac { 4 } { x ^ { 2 } } + 1 , ( - \infty , 0 )$

Find a value of c that makes the function continuous at the given value of x. If it is impossible, state this. - $f ( x ) = \left\{ \begin{array} { c c } ( \sin x ) ^ { x } , & x \neq 0 \\c , & x = 0\end{array} ; \quad x = 0 \right.$

Solve the problem. -Suppose Newton's Method is used with an initial guess $x _ { 0 }$ that lies at a critical point $( a , b ) , b \neq 0$ . What happens to $x _ { 1 }$ and later approximations? Give reasons for your answer.

Solve the problem. -Write down the first four approximations to the solution of the equation $\sin 3 x = x$ using Newton's method with an initial estimate of $x _ { 0 } = 1$ .

Find a value of c that makes the function continuous at the given value of x. If it is impossible, state this. - $f ( x ) = \left\{ \begin{array} { c l } \frac { 4 x - 2 \sin 2 x } { 3 x ^ { 3 } } , & x \neq 0 \\c , & x = 0\end{array} ; x = 0 \right.$

Solve the initial value problem. - $\frac { d y } { d x } = \frac { 1 } { x + 5 } , y ( - 4 ) = 6$

Solve the problem. -Given the velocity and initial position of a body moving along a coordinate line at time $t$ , find the body's position $\mathrm { t }$ $v = - 16 t + 4 , s ( 0 ) = 7$

Functions

Limits and Continuity

Derivatives

Integrals

Applications of Definite Integrals

Integrals and Transcendental Functions

Techniques of Integration

First-Order Differential Equations

Infinite Sequences and Series

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions and Motion in Space

Partial Derivatives

Multiple Integrals

Integrals and Vector Fields

Filters

Exam 5: Applications of Derivatives

Use differentiation to determine whether the integral formula is correct. - ∫(2x−2)4dx=(2x−2)510+C\int ( 2 x - 2 ) ^ { 4 } d x = \frac { ( 2 x - 2 ) ^ { 5 } } { 10 } + C∫(2x−2)4dx=10(2x−2)5​+C

Find an antiderivative of the given function. - cos⁡πx+6sin⁡x6\cos \pi x + 6 \sin \frac { x } { 6 }cosπx+6sin6x​

Solve the problem. -The graphs below show the first and second derivatives of a function y=f(x)y = f ( x )y=f(x) . Select a possible graph fff that passe through the point PPP .

Solve the problem. -Use Newton's method to find the value of xxx for which ln⁡(1/x)=1−x2\ln ( 1 / x ) = 1 - x ^ { 2 }ln(1/x)=1−x2 . Find the answer correct to five decimal places.

Use l'H^opital's rule to find the limit. - lim⁡θ→ −03−3cos⁡θsin⁡4θ\lim _ {\theta \rightarrow \ - 0 } \frac { 3 - 3 \cos \theta } { \sin 4 \theta }limθ→ −0​sin4θ3−3cosθ​

Find the location of the indicated absolute extremum for the function. -

Find the location of the indicated absolute extremum for the function. -

Answer the question. -A marathoner ran the 26.2 mile New York City Marathon in 2.8 hrs. Did the runner ever exceed a speed of 9 miles per hour?

Solve the problem. - Show that g(x)=a+xb2+(a+x)2 is an increasing function of x\text { Show that } g ( x ) = \frac { a + x } { \sqrt { b ^ { 2 } + ( a + x ) ^ { 2 } } } \text { is an increasing function of } x Show that g(x)=b2+(a+x)2​a+x​ is an increasing function of x

Graph the rational function. -\9\begin{array} { c } y = \frac { x ^ { 2 } + x - 42 } { x ^ { 2 } - x - 56 } \\ \end{array}\)

Find an antiderivative of the given function. - x−5+14xx ^ { - 5 } + \frac { 1 } { 4 \sqrt { x } }x−5+4x​1​

Solve the problem. -At about what velocity do you enter the water if you jump from a 15 meter cliff? (Use g=9.8 m/sec2g = 9.8 \mathrm {~m} / \mathrm { sec } ^ { 2 }g=9.8 m/sec2 .)

Show that the function has exactly one zero in the given interval. - f(x)=x3+4x2+1,(−∞,0)f ( x ) = x ^ { 3 } + \frac { 4 } { x ^ { 2 } } + 1 , ( - \infty , 0 )f(x)=x3+x24​+1,(−∞,0)

Find a value of c that makes the function continuous at the given value of x. If it is impossible, state this. - f(x)={(sin⁡x)x,x≠0c,x=0;x=0f ( x ) = \left\{ \begin{array} { c c } ( \sin x ) ^ { x } , & x \neq 0 \\c , & x = 0\end{array} ; \quad x = 0 \right.f(x)={(sinx)x,c,​x=0x=0​;x=0

Solve the problem. -Suppose Newton's Method is used with an initial guess x0x _ { 0 }x0​ that lies at a critical point (a,b),b≠0( a , b ) , b \neq 0(a,b),b=0 . What happens to x1x _ { 1 }x1​ and later approximations? Give reasons for your answer.

Solve the problem. -Write down the first four approximations to the solution of the equation sin⁡3x=x\sin 3 x = xsin3x=x using Newton's method with an initial estimate of x0=1x _ { 0 } = 1x0​=1 .

Solve the initial value problem. - dydx=1x+5,y(−4)=6\frac { d y } { d x } = \frac { 1 } { x + 5 } , y ( - 4 ) = 6dxdy​=x+51​,y(−4)=6

Solve the problem. -Given the velocity and initial position of a body moving along a coordinate line at time ttt , find the body's position t\mathrm { t }t v=−16t+4,s(0)=7v = - 16 t + 4 , s ( 0 ) = 7v=−16t+4,s(0)=7

Functions

Limits and Continuity

Derivatives

Integrals

Applications of Definite Integrals

Integrals and Transcendental Functions

Techniques of Integration

First-Order Differential Equations

Infinite Sequences and Series

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions and Motion in Space

Partial Derivatives

Multiple Integrals

Integrals and Vector Fields

Filters

Use differentiation to determine whether the integral formula is correct. - $\int ( 2 x - 2 ) ^ { 4 } d x = \frac { ( 2 x - 2 ) ^ { 5 } } { 10 } + C$

Find an antiderivative of the given function. - $\cos \pi x + 6 \sin \frac { x } { 6 }$

Solve the problem. -The graphs below show the first and second derivatives of a function $y = f ( x )$ . Select a possible graph $f$ that passe through the point $P$ .

Solve the problem. -Use Newton's method to find the value of $x$ for which $\ln ( 1 / x ) = 1 - x ^ { 2 }$ . Find the answer correct to five decimal places.

Use l'H^opital's rule to find the limit. - $\lim _ {\theta \rightarrow \ - 0 } \frac { 3 - 3 \cos \theta } { \sin 4 \theta }$

Solve the problem. - $\text { Show that } g ( x ) = \frac { a + x } { \sqrt { b ^ { 2 } + ( a + x ) ^ { 2 } } } \text { is an increasing function of } x$

Graph the rational function. -\9\begin{array} { c } y = \frac { x ^ { 2 } + x - 42 } { x ^ { 2 } - x - 56 } \\ \end{array}\) $Graph the rational function. -\9\begin{array} { c } y = \frac { x ^ { 2 } + x - 42 } { x ^ { 2 } - x - 56 } \\ \end{array}\)$

Find an antiderivative of the given function. - $x ^ { - 5 } + \frac { 1 } { 4 \sqrt { x } }$

Solve the problem. -At about what velocity do you enter the water if you jump from a 15 meter cliff? (Use $g = 9.8 \mathrm {~m} / \mathrm { sec } ^ { 2 }$ .)

Show that the function has exactly one zero in the given interval. - $f ( x ) = x ^ { 3 } + \frac { 4 } { x ^ { 2 } } + 1 , ( - \infty , 0 )$

Find a value of c that makes the function continuous at the given value of x. If it is impossible, state this. - $f ( x ) = \left\{ \begin{array} { c c } ( \sin x ) ^ { x } , & x \neq 0 \\c , & x = 0\end{array} ; \quad x = 0 \right.$

Solve the problem. -Suppose Newton's Method is used with an initial guess $x _ { 0 }$ that lies at a critical point $( a , b ) , b \neq 0$ . What happens to $x _ { 1 }$ and later approximations? Give reasons for your answer.

Solve the problem. -Write down the first four approximations to the solution of the equation $\sin 3 x = x$ using Newton's method with an initial estimate of $x _ { 0 } = 1$ .

Solve the initial value problem. - $\frac { d y } { d x } = \frac { 1 } { x + 5 } , y ( - 4 ) = 6$

Solve the problem. -Given the velocity and initial position of a body moving along a coordinate line at time $t$ , find the body's position $\mathrm { t }$ $v = - 16 t + 4 , s ( 0 ) = 7$