Question 1

Estimate the slope of the curve $y = e ^ { x }$at x = 0.

Accepted Answer

A)  1 
B)  0 
C)  e 
D)  $e ^ { x }$ 
E)  none of these 
A)  1 
B)  0 
C)  e 
D)  $e ^ { x }$ 
E)  none of these

Question 2

ln $\left( \frac { 1 } { \mathrm { e } ^ { \mathrm { x } } } \right)$ Enter just a standard polynomial in x.

Accepted Answer

The answer of ln \(\left( \frac { 1 } {...

Question 3

Simplify. 
-( $e ^ { x }$ + $\mathrm { e } ^ { - \mathrm { x } }$ )( $e ^ { x }$ - $\mathrm { e } ^ { - \mathrm { x } }$ )Enter your answer exactly as $e ^ { a } - e ^ { b }$

Accepted Answer

The answer of Simplify. 
-( $e ^ { x }$...

Question 4

Which of the following properties are true of the graph of y = 10 $\mathrm { e } ^ { 2 x }$ ? 
(I) It is concave up.
(II) The y-intercept is (0, 2).
(III) It has a minimum at x = 0.
(IV) y is positive for x ≥ 0 and negative for x < 0.

Accepted Answer

A)  I only 
B)  I and III 
C)  I and II 
D)  III and IV 
A)  I only 
B)  I and III 
C)  I and II 
D)  III and IV

Question 5

Solve for x. 
-$\mathrm { e } ^ { \left( \mathrm { x } ^ { 2 } + 9 \right) }$ ∙ $e ^ { ( 6 x ) }$ = 1

Accepted Answer

A)  x = 0 
B)  x = ln $\frac { 1 } { 2 }$ 
C)  x = -3 
D)  x = ± $\sqrt { 3 }$ 
A)  x = 0 
B)  x = ln $\frac { 1 } { 2 }$ 
C)  x = -3 
D)  x = ± $\sqrt { 3 }$

Question 6

Use logarithmic differentiation to differentiate.
-ln $\left[ \frac { \sqrt { \mathrm { xe } ^ { x } } } { \mathrm { x } ^ { 2 } + 1 } \right]$ at x = 1
Enter just a reduced fraction of form $\frac { a } { b }$ .

Accepted Answer

The answer of Use logarithmic differentiation to differentiate.
-ln \(\left[ \frac...

Question 7

Choose the one alternative that best completes the statement or answers the question. 
-$16 ( \mathrm { p } / 2 )$ ∙ $\left( \frac { 1 } { 2 } \right) ^ { p }$ ∙ $\left( \frac { 1 } { 2 } \right) ^ { p }$

Accepted Answer

A)  $4 \mathrm { P }$ 
B)  $\left( \frac { 4 } { p } \right) ^ { ( 3 / 2 ) p + 1 }$ 
C)  1 
D)  cannot be simplified 
A)  $4 \mathrm { P }$ 
B)  $\left( \frac { 4 } { p } \right) ^ { ( 3 / 2 ) p + 1 }$ 
C)  1 
D)  cannot be simplified

Question 8

Choose the one alternative that best completes the statement or answers the question. 
-( $t ^ { 2 }$ $) ^ { x }$ ∙ ( $t^4$ $) ^ { x }$ ∙ ( $t ^{1 / 3}$ $) ^ { x }$

Accepted Answer

A)  ( $t ^{19 / 3 x}$ $) ^ { 3 x }$ 
B)  $t ^{( 8 / 3 ) x}$ 
C)  $\left((t 19 / 3)^{x}\right.$ 
D)  ( $t ^ { 8 }$ $) ^ { x / 3 }$ 
A)  ( $t ^{19 / 3 x}$ $) ^ { 3 x }$ 
B)  $t ^{( 8 / 3 ) x}$ 
C)  $\left((t 19 / 3)^{x}\right.$ 
D)  ( $t ^ { 8 }$ $) ^ { x / 3 }$

Question 9

Solve for x. 
-ln(1 + $x ^ { 2 }$ ) = 2.
Enter your answer exactly as just ± $\sqrt { \mathrm { e } ^ { a } \pm b }$ (a, b integers).

Accepted Answer

The answer of Solve for x. 
-ln(1 + \(x ^...

Question 10

Differentiate. 
-$e ^ { 3 x }$

Accepted Answer

A)  3x 
B)  $e ^ { 3 x }$ 
C)  3 $e ^ { 3 x }$ 
D)  $\frac { 1 } { 3 }$ $e ^ { 3 x }$ 
A)  3x 
B)  $e ^ { 3 x }$ 
C)  3 $e ^ { 3 x }$ 
D)  $\frac { 1 } { 3 }$ $e ^ { 3 x }$

Question 11

Differentiate.
-(6 $\mathrm { e } ^ { 2 x }$ - x $) ^ { 3 }$

Accepted Answer

A)  3(12 $x e ^ { 2 x - 1 }$ - 1 $)^2$ 
B)  3(6 $\mathrm { e } ^ { 2 x }$ - x $)^2$ (12 $\mathrm { e } ^ { 2 x }$ - 1) 
C)  3(12 $e ^ { x }$ - 1 $)^2$ 
D)  3(6 $\mathrm { e } ^ { 2 x }$ - x $)^2$ (12 $\mathrm { e } ^ { 2 x }$ ) 
A)  3(12 $x e ^ { 2 x - 1 }$ - 1 $)^2$ 
B)  3(6 $\mathrm { e } ^ { 2 x }$ - x $)^2$ (12 $\mathrm { e } ^ { 2 x }$ - 1) 
C)  3(12 $e ^ { x }$ - 1 $)^2$ 
D)  3(6 $\mathrm { e } ^ { 2 x }$ - x $)^2$ (12 $\mathrm { e } ^ { 2 x }$ )

Question 12

Differentiate. 
-ln $\frac { x + 2 } { x - 1 }$

Accepted Answer

A)  $\frac { 3 x } { ( x + 2 ) ( x - 1 ) }$ 
B)  $\frac { 1 } { x + 2 }$ - $\frac { 1 } { x - 1 }$ 
C)  $\frac { ( x - 1 ) } { ( x + 2 ) ( x + 3 ) }$ 
D)  $\frac { x - 1 } { x + 2 }$ 
E)  none of these 
A)  $\frac { 3 x } { ( x + 2 ) ( x - 1 ) }$ 
B)  $\frac { 1 } { x + 2 }$ - $\frac { 1 } { x - 1 }$ 
C)  $\frac { ( x - 1 ) } { ( x + 2 ) ( x + 3 ) }$ 
D)  $\frac { x - 1 } { x + 2 }$ 
E)  none of these

Question 13

Solve for x. 
-2 - ln(x + 3) = ln 4

Accepted Answer

A)  x = ln 4 - 1 
B)  x = 2e 
C)  x = -3 
D)  x = $\frac { 1 } { 4 }$ $e ^ { 2 }$ - 3 
A)  x = ln 4 - 1 
B)  x = 2e 
C)  x = -3 
D)  x = $\frac { 1 } { 4 }$ $e ^ { 2 }$ - 3

Question 14

Solve for x: $3 ^ { 5 } x$ ∙ $3 x ^ { 2 }$ ∙ $3 ^ { 3 }$ = $3 - 3$ .
Enter your answer exactly as x = a, b ( a < b).

Accepted Answer

The answer of Solve for x: \(3 ^ { 5...

Question 15

The expression may be factored as shown. Find the missing factor. $5 ^ { 2 } + h$ = 25(   )
Enter your answer as $5 ^a$ .

Accepted Answer

The answer of The expression may be factored as shown....

Question 16

Differentiate. 
-(ln( $x ^ { 2 }$ + 2) $) ^ { 3 }$

Accepted Answer

A)  e $\left( \frac { 1 } { x ^ { 2 } + 2 } \right) ^ { 2 }$ ∙ 2x 
B)  3(ln(2x) $) ^ { 2 }$ 
C)  $\frac { 6 x } { x ^ { 2 } + 2 }$ (ln( $x ^ { 2 }$ + 2) $)2$ 
D)  $\frac { 1 } { \left( \ln \left( x ^ { 2 } + 2 \right) \right) ^ { 3 } }$ ∙ 2x 
A)  e $\left( \frac { 1 } { x ^ { 2 } + 2 } \right) ^ { 2 }$ ∙ 2x 
B)  3(ln(2x) $) ^ { 2 }$ 
C)  $\frac { 6 x } { x ^ { 2 } + 2 }$ (ln( $x ^ { 2 }$ + 2) $)2$ 
D)  $\frac { 1 } { \left( \ln \left( x ^ { 2 } + 2 \right) \right) ^ { 3 } }$ ∙ 2x

Question 17

f(x) = $\frac { \ln x } { \mathrm { e } ^ { \mathrm { x } } }$ at x = 1 Enter your answer as just $e ^ { a }$ .

Accepted Answer

The answer of f(x) = \(\frac { \ln x }...

Question 18

f(x) = $\frac { e ^ { x } + 1 } { e ^ { x } - 1 }$ Enter your answer exactly as just $\frac { \mathrm { P } \left( \mathrm { e } ^ { \mathrm { x } } \right) } { \left( \mathrm { Q } \left( \mathrm { e } ^ { \mathrm { x } } \right) \right) ^ { \mathrm { n } } }$ where P and Q are polynomials in $e ^ { x }$ in standard form.

Accepted Answer

The answer of f(x) = \(\frac { e ^ {...

Question 19

Solve for x. 
-ln(x + 1) = 2 + ln x

Accepted Answer

A)  $e ^ { 2 }$ - 1 
B)  e + 1 
C)  $\frac { 1 } { e ^ { 2 } - 1 }$ 
D)  e - 1 
E)  none of these 
A)  $e ^ { 2 }$ - 1 
B)  e + 1 
C)  $\frac { 1 } { e ^ { 2 } - 1 }$ 
D)  e - 1 
E)  none of these

Question 20

ln $x ^ { 2 }$ + (ln x $)^2$ = 0
Enter your answer exactly as x = a, b (a < b).

Accepted Answer

The answer of ln $x ^ { 2 }$ +...

Estimate the slope of the curve $y = e ^ { x }$ at x = 0.

ln $\left( \frac { 1 } { \mathrm { e } ^ { \mathrm { x } } } \right)$ Enter just a standard polynomial in x.

Simplify. -( $e ^ { x }$ + $\mathrm { e } ^ { - \mathrm { x } }$ )( $e ^ { x }$ - $\mathrm { e } ^ { - \mathrm { x } }$ )Enter your answer exactly as $e ^ { a } - e ^ { b }$

Which of the following properties are true of the graph of y = 10 $\mathrm { e } ^ { 2 x }$ ? (I) It is concave up. (II) The y-intercept is (0, 2). (III) It has a minimum at x = 0. (IV) y is positive for x ≥ 0 and negative for x < 0.

Solve for x. - $\mathrm { e } ^ { \left( \mathrm { x } ^ { 2 } + 9 \right) }$ ∙ $e ^ { ( 6 x ) }$ = 1

Use logarithmic differentiation to differentiate. -ln $\left[ \frac { \sqrt { \mathrm { xe } ^ { x } } } { \mathrm { x } ^ { 2 } + 1 } \right]$ at x = 1 Enter just a reduced fraction of form $\frac { a } { b }$ .

Choose the one alternative that best completes the statement or answers the question. - $16 ( \mathrm { p } / 2 )$ ∙ $\left( \frac { 1 } { 2 } \right) ^ { p }$ ∙ $\left( \frac { 1 } { 2 } \right) ^ { p }$

Choose the one alternative that best completes the statement or answers the question. -( $t ^ { 2 }$ $) ^ { x }$ ∙ ( $t^4$ $) ^ { x }$ ∙ ( $t ^{1 / 3}$ $) ^ { x }$

Solve for x. -ln(1 + $x ^ { 2 }$ ) = 2. Enter your answer exactly as just ± $\sqrt { \mathrm { e } ^ { a } \pm b }$ (a, b integers).

Differentiate. - $e ^ { 3 x }$

Differentiate. -(6 $\mathrm { e } ^ { 2 x }$ - x $) ^ { 3 }$

Differentiate. -ln $\frac { x + 2 } { x - 1 }$

Solve for x. -2 - ln(x + 3) = ln 4

Solve for x: $3 ^ { 5 } x$ ∙ $3 x ^ { 2 }$ ∙ $3 ^ { 3 }$ = $3 - 3$ . Enter your answer exactly as x = a, b ( a < b).

The expression may be factored as shown. Find the missing factor. $5 ^ { 2 } + h$ = 25( $The expression may be factored as shown. Find the missing factor. 5 ^ { 2 } + h = 25( ) Enter your answer as 5 ^a .$ ) Enter your answer as $5 ^a$ .

Differentiate. -(ln( $x ^ { 2 }$ + 2) $) ^ { 3 }$

f(x) = $\frac { \ln x } { \mathrm { e } ^ { \mathrm { x } } }$ at x = 1 Enter your answer as just $e ^ { a }$ .

Solve for x. -ln(x + 1) = 2 + ln x

ln $x ^ { 2 }$ + (ln x $)^2$ = 0 Enter your answer exactly as x = a, b (a < b).

The Derivative

Applications of the Derivative

Techniques of Differentiation

Applications of the Exponential and Natural Logarithm Functions

The Definite Integral

Functions of Several Variables

The Trigonometric Functions

Techniques of Integration

Differential Equations

Taylor Polynomials and Infinite Series

Probability and Calculus

Filters

Exam 4: Logarithm Functions

Estimate the slope of the curve y=exy = e ^ { x }y=ex at x = 0.

ln (1ex)\left( \frac { 1 } { \mathrm { e } ^ { \mathrm { x } } } \right)(ex1​) Enter just a standard polynomial in x.

Simplify. -( exe ^ { x }ex + e−x\mathrm { e } ^ { - \mathrm { x } }e−x )( exe ^ { x }ex - e−x\mathrm { e } ^ { - \mathrm { x } }e−x )Enter your answer exactly as ea−ebe ^ { a } - e ^ { b }ea−eb

Which of the following properties are true of the graph of y = 10 e2x\mathrm { e } ^ { 2 x }e2x ? (I) It is concave up. (II) The y-intercept is (0, 2). (III) It has a minimum at x = 0. (IV) y is positive for x ≥ 0 and negative for x < 0.

Solve for x. - e(x2+9)\mathrm { e } ^ { \left( \mathrm { x } ^ { 2 } + 9 \right) }e(x2+9) ∙ e(6x)e ^ { ( 6 x ) }e(6x) = 1

Use logarithmic differentiation to differentiate. -ln [xexx2+1]\left[ \frac { \sqrt { \mathrm { xe } ^ { x } } } { \mathrm { x } ^ { 2 } + 1 } \right][x2+1xex​​] at x = 1 Enter just a reduced fraction of form ab\frac { a } { b }ba​ .

Choose the one alternative that best completes the statement or answers the question. - 16(p/2)16 ( \mathrm { p } / 2 )16(p/2) ∙ (12)p\left( \frac { 1 } { 2 } \right) ^ { p }(21​)p ∙ (12)p\left( \frac { 1 } { 2 } \right) ^ { p }(21​)p

Choose the one alternative that best completes the statement or answers the question. -( t2t ^ { 2 }t2 )x) ^ { x })x ∙ ( t4t^4t4 )x) ^ { x })x ∙ ( t1/3t ^{1 / 3}t1/3 )x) ^ { x })x

Solve for x. -ln(1 + x2x ^ { 2 }x2 ) = 2. Enter your answer exactly as just ± ea±b\sqrt { \mathrm { e } ^ { a } \pm b }ea±b​ (a, b integers).

Differentiate. - e3xe ^ { 3 x }e3x

Differentiate. -(6 e2x\mathrm { e } ^ { 2 x }e2x - x )3) ^ { 3 })3

Differentiate. -ln x+2x−1\frac { x + 2 } { x - 1 }x−1x+2​

Solve for x. -2 - ln(x + 3) = ln 4

Solve for x: 35x3 ^ { 5 } x35x ∙ 3x23 x ^ { 2 }3x2 ∙ 333 ^ { 3 }33 = 3−33 - 33−3 . Enter your answer exactly as x = a, b ( a < b).

The expression may be factored as shown. Find the missing factor. 52+h5 ^ { 2 } + h52+h = 25( ) Enter your answer as 5a5 ^a5a .

Differentiate. -(ln( x2x ^ { 2 }x2 + 2) )3) ^ { 3 })3

f(x) = ln⁡xex\frac { \ln x } { \mathrm { e } ^ { \mathrm { x } } }exlnx​ at x = 1 Enter your answer as just eae ^ { a }ea .

Solve for x. -ln(x + 1) = 2 + ln x

ln x2x ^ { 2 }x2 + (ln x )2)^2)2 = 0 Enter your answer exactly as x = a, b (a < b).

The Derivative

Applications of the Derivative

Techniques of Differentiation

Applications of the Exponential and Natural Logarithm Functions

The Definite Integral

Functions of Several Variables

The Trigonometric Functions

Techniques of Integration

Differential Equations

Taylor Polynomials and Infinite Series

Probability and Calculus

Filters

Estimate the slope of the curve $y = e ^ { x }$ at x = 0.

ln $\left( \frac { 1 } { \mathrm { e } ^ { \mathrm { x } } } \right)$ Enter just a standard polynomial in x.

Simplify. -( $e ^ { x }$ + $\mathrm { e } ^ { - \mathrm { x } }$ )( $e ^ { x }$ - $\mathrm { e } ^ { - \mathrm { x } }$ )Enter your answer exactly as $e ^ { a } - e ^ { b }$

Which of the following properties are true of the graph of y = 10 $\mathrm { e } ^ { 2 x }$ ? (I) It is concave up. (II) The y-intercept is (0, 2). (III) It has a minimum at x = 0. (IV) y is positive for x ≥ 0 and negative for x < 0.

Solve for x. - $\mathrm { e } ^ { \left( \mathrm { x } ^ { 2 } + 9 \right) }$ ∙ $e ^ { ( 6 x ) }$ = 1

Use logarithmic differentiation to differentiate. -ln $\left[ \frac { \sqrt { \mathrm { xe } ^ { x } } } { \mathrm { x } ^ { 2 } + 1 } \right]$ at x = 1 Enter just a reduced fraction of form $\frac { a } { b }$ .

Choose the one alternative that best completes the statement or answers the question. - $16 ( \mathrm { p } / 2 )$ ∙ $\left( \frac { 1 } { 2 } \right) ^ { p }$ ∙ $\left( \frac { 1 } { 2 } \right) ^ { p }$

Choose the one alternative that best completes the statement or answers the question. -( $t ^ { 2 }$ $) ^ { x }$ ∙ ( $t^4$ $) ^ { x }$ ∙ ( $t ^{1 / 3}$ $) ^ { x }$

Solve for x. -ln(1 + $x ^ { 2 }$ ) = 2. Enter your answer exactly as just ± $\sqrt { \mathrm { e } ^ { a } \pm b }$ (a, b integers).

Differentiate. - $e ^ { 3 x }$

Differentiate. -(6 $\mathrm { e } ^ { 2 x }$ - x $) ^ { 3 }$

Differentiate. -ln $\frac { x + 2 } { x - 1 }$

Solve for x: $3 ^ { 5 } x$ ∙ $3 x ^ { 2 }$ ∙ $3 ^ { 3 }$ = $3 - 3$ . Enter your answer exactly as x = a, b ( a < b).

The expression may be factored as shown. Find the missing factor. $5 ^ { 2 } + h$ = 25( $The expression may be factored as shown. Find the missing factor. 5 ^ { 2 } + h = 25( ) Enter your answer as 5 ^a .$ ) Enter your answer as $5 ^a$ .

Differentiate. -(ln( $x ^ { 2 }$ + 2) $) ^ { 3 }$

f(x) = $\frac { \ln x } { \mathrm { e } ^ { \mathrm { x } } }$ at x = 1 Enter your answer as just $e ^ { a }$ .

ln $x ^ { 2 }$ + (ln x $)^2$ = 0 Enter your answer exactly as x = a, b (a < b).