Question 1

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

Accepted Answer

A)  2 $e ^ { x }$ + 2 $\frac { 1 } { \ln \left( x ^ { \mathrm { e } } \right) }$ ∙ $x ^ { e }$ 
B)  2x $e ^ { x ^ { 2 } }$ + 2e $\frac { 1 } { x }$ 
C)  $x ^ { 2 }$ $e ^ { x ^ { 2 } - 1 }$ + 2e $\frac { 1 } { \mathrm { x } ^ { \mathrm { e } } }$ ∙ $x ^ { e }$ - 1 
D)  none of these 
A)  2 $e ^ { x }$ + 2 $\frac { 1 } { \ln \left( x ^ { \mathrm { e } } \right) }$ ∙ $x ^ { e }$ 
B)  2x $e ^ { x ^ { 2 } }$ + 2e $\frac { 1 } { x }$ 
C)  $x ^ { 2 }$ $e ^ { x ^ { 2 } - 1 }$ + 2e $\frac { 1 } { \mathrm { x } ^ { \mathrm { e } } }$ ∙ $x ^ { e }$ - 1 
D)  none of these B

Question 2

Find an equation of the tangent line to the graph of y = $x ^ { 3 }$ ln(-2x) at x = -1.

Accepted Answer

A)  y = (x + ln 2) - 1 
B)  y = (1 + 3 ln 2)(x + 1) - ln 2 
C)  y - 1 = 4(x - ln 2) 
D)  y + ln 2 = 4(x - 1) 
A)  y = (x + ln 2) - 1 
B)  y = (1 + 3 ln 2)(x + 1) - ln 2 
C)  y - 1 = 4(x - ln 2) 
D)  y + ln 2 = 4(x - 1) B

Question 3

Is this the graph of y = 2x - ln $x ^ { 2 }$ , x > 0?

Accepted Answer

A) True 
 B)False  True

Question 4

If ( $e ^ { x }$ $)^2$ ∙ $\mathrm { e } ^ { 2 x }$ ∙ e = $\frac { 1 } { e ^ { 2 } }$ , find x.

Accepted Answer

A)  -2 
B)  -1 or -2 
C)  - $\frac { 3 } { 4 }$ 
D)  - $\frac { 2 } { 5 }$
Differentiate. 
A)  -2 
B)  -1 or -2 
C)  - $\frac { 3 } { 4 }$ 
D)  - $\frac { 2 } { 5 }$
Differentiate.

Question 5

Simplify.
-$e ^ { 2 } \ln 5$ + ln( $e ^ { x }$ ∙ $\mathrm { e } ^ { 4 }$ )

Accepted Answer

A)  10 + x ∙ $\mathrm { e } ^ { 4 }$ 
B)  29 + x 
C)  25 + 4x 
D)  10 + 4x 
A)  10 + x ∙ $\mathrm { e } ^ { 4 }$ 
B)  29 + x 
C)  25 + 4x 
D)  10 + 4x

Question 6

Let y = $\mathrm { e } ^ { \mathrm { e } ^ { 2 x + 1 } }$ . What is $\frac { d y } { d x }$ ?

Accepted Answer

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

Question 7

Find the first and second derivatives of f(x) = $$\frac { 1 - 2 x } { e ^ { x } }$$ .

Accepted Answer

A)  f'(x) = 2x $e ^ { x }$ - 3 $e ^ { x }$, F''(x) = 2x $e ^ { x }$ - $e ^ { x }$ 
B)  f'(x) = 4x $e ^ { x }$ - 2 $e ^ { x }$, F''(x) = 4x $e ^ { x }$ + 2 $e ^ { x }$ 
C)  f'(x) = $\frac { 3 - 2 x } { e ^ { x } }$, F''(x) = $\frac { 2 x - 5 } { e ^ { x } }$ 
D)  f'(x) = $\frac { 2 x - 3 } { e ^ { x } }$, F''(x) = $\frac { 5 - 2 x } { e ^ { x } }$ 
A)  f'(x) = 2x $e ^ { x }$ - 3 $e ^ { x }$, F''(x) = 2x $e ^ { x }$ - $e ^ { x }$ 
B)  f'(x) = 4x $e ^ { x }$ - 2 $e ^ { x }$, F''(x) = 4x $e ^ { x }$ + 2 $e ^ { x }$ 
C)  f'(x) = $\frac { 3 - 2 x } { e ^ { x } }$, F''(x) = $\frac { 2 x - 5 } { e ^ { x } }$ 
D)  f'(x) = $\frac { 2 x - 3 } { e ^ { x } }$, F''(x) = $\frac { 5 - 2 x } { e ^ { x } }$

Question 8

If $\left( \frac { 1 } { 9 } \right) ^ { 3 x + 4 }$ = 81, find x.

Accepted Answer

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

Question 9

Use logarithmic differentiation to differentiate.
-$( 3 x + 1 ) ^ { 5 }$ $( 2 x - 1 ) ^ { - 2 }$ $( x + 3 ) ^ { 4 }$ at x = 1
Enter your answer exactly as just 3 ∙ $4 ^ { a }$ .

Accepted Answer

The answer of Use logarithmic differentiation to differentiate.
-\(( 3 x...

Question 10

Solve for x. 
-ln $x ^ { 3 }$ + 3 ln x = 0

Accepted Answer

A)  x = 1 
B)  x = 0, ± $\sqrt { 3 }$ 
C)  x = $\mathrm { e } ^ { 3 }$ 
D)  x = $\sqrt { 3 }$ 
A)  x = 1 
B)  x = 0, ± $\sqrt { 3 }$ 
C)  x = $\mathrm { e } ^ { 3 }$ 
D)  x = $\sqrt { 3 }$

Question 11

Simplify. 
-$\sqrt { \frac { e ^ { ( 1 / 2 ) x } \cdot e ^ { ( 3 / 2 ) x } } { e ^ { - 2 x } } }$

Accepted Answer

A)  $\frac { 1 } { e ^ { 2 x } }$ 
B)  $e ^ { - 6 x }$ 
C)  $e ^ { ( 1 / 2 ) x }$ 
D)  $e ^ { - 10 x }$ 
E)  none of these 
A)  $\frac { 1 } { e ^ { 2 x } }$ 
B)  $e ^ { - 6 x }$ 
C)  $e ^ { ( 1 / 2 ) x }$ 
D)  $e ^ { - 10 x }$ 
E)  none of these

Question 12

Solve for x. 
-$\mathrm { e } ^ { 1 + x }$ - 2 = 4
Enter your answer exactly as just a ± ln b (a, b integers).

Accepted Answer

The answer of Solve for x. 
-\(\mathrm { e }...

Question 13

Solve for x: $7^{ - 5}$ ∙ 49 ∙ $7 x ^ { 2 }$ ∙ $49 x$ = 1.
Enter your answer exactly as x = a, b (a < b).

Accepted Answer

The answer of Solve for x: $7^{ - 5}$ ∙...

Question 14

If $\mathrm { e } ^ { - \mathrm { x } }$ = 6, write x in terms of the natural logarithm.

Accepted Answer

A)  ln 6 
B)  -ln 6 
C)  ln -6 
D)  6 
A)  ln 6 
B)  -ln 6 
C)  ln -6 
D)  6

Question 15

ln $e ^ { 1 / x }$ Enter your answer exactly in the form $a^ b$ where b is an integer.

Accepted Answer

The answer of ln \(e ^ { 1 / x...

Question 16

$e ^ { \ln 2 x }$ Enter just a standard polynomial in x.

Accepted Answer

The answer of $e ^ { \ln 2 x }$...

Question 17

2 ln x + 3 ln x = 4
Enter your answer exactly as just $\mathrm { e } ^ { \mathrm { a } / \mathrm { b } }$ (a, b integers).

Accepted Answer

The answer of 2 ln x + 3 ln x...

Question 18

$$\text { If } 5 ^ { t } .5 ^ { 4 t } .5 ^ { - 2 t } = 25^{ ( - 1 / 2 ) t + 1} \text {, find } t \text {. }$$

Accepted Answer

A)  t = $\frac { 4 } { 13 }$ 
B)  t = - $\frac { 1 } { 2 }$ 
C)  t = $\frac { 1 } { 2 }$ 
D)  t = $\frac { 2 } { 7 }$ 
A)  t = $\frac { 4 } { 13 }$ 
B)  t = - $\frac { 1 } { 2 }$ 
C)  t = $\frac { 1 } { 2 }$ 
D)  t = $\frac { 2 } { 7 }$

Question 19

4 $e ^ { 3 x + 2 }$ = 20
Enter your answer exactly as x = $\frac { \ln a - b } { c }$ .

Accepted Answer

The answer of 4 \(e ^ { 3 x +...

Question 20

$\mathrm { e } ^ { ( \ln x ) ^ { 2 } }$ Enter your answer exactly as $e ^ { ( \ln a ) ^ { b } \frac { c \ln d } { f } }$ .

Accepted Answer

The answer of \(\mathrm { e } ^ { (...

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

Find an equation of the tangent line to the graph of y = $x ^ { 3 }$ ln(-2x) at x = -1.

Is this the graph of y = 2x - ln $x ^ { 2 }$ , x > 0? $Is this the graph of y = 2x - ln x ^ { 2 } , x > 0?$

If ( $e ^ { x }$ $)^2$ ∙ $\mathrm { e } ^ { 2 x }$ ∙ e = $\frac { 1 } { e ^ { 2 } }$ , find x.

Simplify. - $e ^ { 2 } \ln 5$ + ln( $e ^ { x }$ ∙ $\mathrm { e } ^ { 4 }$ )

Let y = $\mathrm { e } ^ { \mathrm { e } ^ { 2 x + 1 } }$ . What is $\frac { d y } { d x }$ ?

Find the first and second derivatives of f(x) = $\frac { 1 - 2 x } { e ^ { x } }$ .

If $\left( \frac { 1 } { 9 } \right) ^ { 3 x + 4 }$ = 81, find x.

Use logarithmic differentiation to differentiate. - $( 3 x + 1 ) ^ { 5 }$ $( 2 x - 1 ) ^ { - 2 }$ $( x + 3 ) ^ { 4 }$ at x = 1 Enter your answer exactly as just 3 ∙ $4 ^ { a }$ .

Solve for x. -ln $x ^ { 3 }$ + 3 ln x = 0

Simplify. - $\sqrt { \frac { e ^ { ( 1 / 2 ) x } \cdot e ^ { ( 3 / 2 ) x } } { e ^ { - 2 x } } }$

Solve for x. - $\mathrm { e } ^ { 1 + x }$ - 2 = 4 Enter your answer exactly as just a ± ln b (a, b integers).

Solve for x: $7^{ - 5}$ ∙ 49 ∙ $7 x ^ { 2 }$ ∙ $49 x$ = 1. Enter your answer exactly as x = a, b (a < b).

If $\mathrm { e } ^ { - \mathrm { x } }$ = 6, write x in terms of the natural logarithm.

ln $e ^ { 1 / x }$ Enter your answer exactly in the form $a^ b$ where b is an integer.

$e ^ { \ln 2 x }$ Enter just a standard polynomial in x.

2 ln x + 3 ln x = 4 Enter your answer exactly as just $\mathrm { e } ^ { \mathrm { a } / \mathrm { b } }$ (a, b integers).

$\text { If } 5 ^ { t } .5 ^ { 4 t } .5 ^ { - 2 t } = 25^{ ( - 1 / 2 ) t + 1} \text {, find } t \text {. }$

4 $e ^ { 3 x + 2 }$ = 20 Enter your answer exactly as x = $\frac { \ln a - b } { c }$ .

$\mathrm { e } ^ { ( \ln x ) ^ { 2 } }$ Enter your answer exactly as $e ^ { ( \ln a ) ^ { b } \frac { c \ln d } { f } }$ .

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

Differentiate. - ex2e ^ { x ^ { 2 } }ex2 + 2 ln( xex ^ { e }xe )

Find an equation of the tangent line to the graph of y = x3x ^ { 3 }x3 ln(-2x) at x = -1.

Is this the graph of y = 2x - ln x2x ^ { 2 }x2 , x > 0?

If ( exe ^ { x }ex )2)^2)2 ∙ e2x\mathrm { e } ^ { 2 x }e2x ∙ e = 1e2\frac { 1 } { e ^ { 2 } }e21​ , find x.

Simplify. - e2ln⁡5e ^ { 2 } \ln 5e2ln5 + ln( exe ^ { x }ex ∙ e4\mathrm { e } ^ { 4 }e4 )

Let y = ee2x+1\mathrm { e } ^ { \mathrm { e } ^ { 2 x + 1 } }ee2x+1 . What is dydx\frac { d y } { d x }dxdy​ ?

Find the first and second derivatives of f(x) = 1−2xex\frac { 1 - 2 x } { e ^ { x } }ex1−2x​ .

If (19)3x+4\left( \frac { 1 } { 9 } \right) ^ { 3 x + 4 }(91​)3x+4 = 81, find x.

Use logarithmic differentiation to differentiate. - (3x+1)5( 3 x + 1 ) ^ { 5 }(3x+1)5 (2x−1)−2( 2 x - 1 ) ^ { - 2 }(2x−1)−2 (x+3)4( x + 3 ) ^ { 4 }(x+3)4 at x = 1 Enter your answer exactly as just 3 ∙ 4a4 ^ { a }4a .

Solve for x. -ln x3x ^ { 3 }x3 + 3 ln x = 0

Simplify. - e(1/2)x⋅e(3/2)xe−2x\sqrt { \frac { e ^ { ( 1 / 2 ) x } \cdot e ^ { ( 3 / 2 ) x } } { e ^ { - 2 x } } }e−2xe(1/2)x⋅e(3/2)x​​

Solve for x. - e1+x\mathrm { e } ^ { 1 + x }e1+x - 2 = 4 Enter your answer exactly as just a ± ln b (a, b integers).

Solve for x: 7−57^{ - 5}7−5 ∙ 49 ∙ 7x27 x ^ { 2 }7x2 ∙ 49x49 x49x = 1. Enter your answer exactly as x = a, b (a < b).

If e−x\mathrm { e } ^ { - \mathrm { x } }e−x = 6, write x in terms of the natural logarithm.

ln e1/xe ^ { 1 / x }e1/x Enter your answer exactly in the form aba^ bab where b is an integer.

eln⁡2xe ^ { \ln 2 x }eln2x Enter just a standard polynomial in x.

2 ln x + 3 ln x = 4 Enter your answer exactly as just ea/b\mathrm { e } ^ { \mathrm { a } / \mathrm { b } }ea/b (a, b integers).

If 5t.54t.5−2t=25(−1/2)t+1, find t. \text { If } 5 ^ { t } .5 ^ { 4 t } .5 ^ { - 2 t } = 25^{ ( - 1 / 2 ) t + 1} \text {, find } t \text {. } If 5t.54t.5−2t=25(−1/2)t+1, find t.

4 e3x+2e ^ { 3 x + 2 }e3x+2 = 20 Enter your answer exactly as x = ln⁡a−bc\frac { \ln a - b } { c }clna−b​ .

e(ln⁡x)2\mathrm { e } ^ { ( \ln x ) ^ { 2 } }e(lnx)2 Enter your answer exactly as e(ln⁡a)bcln⁡dfe ^ { ( \ln a ) ^ { b } \frac { c \ln d } { f } }e(lna)bfclnd​ .

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

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

Find an equation of the tangent line to the graph of y = $x ^ { 3 }$ ln(-2x) at x = -1.

Is this the graph of y = 2x - ln $x ^ { 2 }$ , x > 0? $Is this the graph of y = 2x - ln x ^ { 2 } , x > 0?$

If ( $e ^ { x }$ $)^2$ ∙ $\mathrm { e } ^ { 2 x }$ ∙ e = $\frac { 1 } { e ^ { 2 } }$ , find x.

Simplify. - $e ^ { 2 } \ln 5$ + ln( $e ^ { x }$ ∙ $\mathrm { e } ^ { 4 }$ )

Let y = $\mathrm { e } ^ { \mathrm { e } ^ { 2 x + 1 } }$ . What is $\frac { d y } { d x }$ ?

Find the first and second derivatives of f(x) = $\frac { 1 - 2 x } { e ^ { x } }$ .

If $\left( \frac { 1 } { 9 } \right) ^ { 3 x + 4 }$ = 81, find x.

Use logarithmic differentiation to differentiate. - $( 3 x + 1 ) ^ { 5 }$ $( 2 x - 1 ) ^ { - 2 }$ $( x + 3 ) ^ { 4 }$ at x = 1 Enter your answer exactly as just 3 ∙ $4 ^ { a }$ .

Solve for x. -ln $x ^ { 3 }$ + 3 ln x = 0

Simplify. - $\sqrt { \frac { e ^ { ( 1 / 2 ) x } \cdot e ^ { ( 3 / 2 ) x } } { e ^ { - 2 x } } }$

Solve for x. - $\mathrm { e } ^ { 1 + x }$ - 2 = 4 Enter your answer exactly as just a ± ln b (a, b integers).

Solve for x: $7^{ - 5}$ ∙ 49 ∙ $7 x ^ { 2 }$ ∙ $49 x$ = 1. Enter your answer exactly as x = a, b (a < b).

If $\mathrm { e } ^ { - \mathrm { x } }$ = 6, write x in terms of the natural logarithm.

ln $e ^ { 1 / x }$ Enter your answer exactly in the form $a^ b$ where b is an integer.

$e ^ { \ln 2 x }$ Enter just a standard polynomial in x.

2 ln x + 3 ln x = 4 Enter your answer exactly as just $\mathrm { e } ^ { \mathrm { a } / \mathrm { b } }$ (a, b integers).

$\text { If } 5 ^ { t } .5 ^ { 4 t } .5 ^ { - 2 t } = 25^{ ( - 1 / 2 ) t + 1} \text {, find } t \text {. }$

4 $e ^ { 3 x + 2 }$ = 20 Enter your answer exactly as x = $\frac { \ln a - b } { c }$ .

$\mathrm { e } ^ { ( \ln x ) ^ { 2 } }$ Enter your answer exactly as $e ^ { ( \ln a ) ^ { b } \frac { c \ln d } { f } }$ .