Question 1

Solve for x. 
-$e ^ { 4 x ^ { 2 } }$ + $e ^ { ( 2 x ) ^ { 2 } }$ = 6

Accepted Answer

A)  x = ± $\frac { \sqrt { \ln 3 } } { 2 }$ 
B)  x = $\frac { 1 } { 2 }$ ln $\frac { 1 } { 2 }$ 
C)  x = ± $\frac { 1 } { 4 }$ $\sqrt { \ln 6 }$ 
D)  x = ± $\frac { 1 } { 2 }$ $\sqrt { \ln 6 }$ 
A)  x = ± $\frac { \sqrt { \ln 3 } } { 2 }$ 
B)  x = $\frac { 1 } { 2 }$ ln $\frac { 1 } { 2 }$ 
C)  x = ± $\frac { 1 } { 4 }$ $\sqrt { \ln 6 }$ 
D)  x = ± $\frac { 1 } { 2 }$ $\sqrt { \ln 6 }$

Question 2

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

Accepted Answer

A)  (ln + 1)( $x ^ { 2 }$ - 2) 
B)  ln( $x ^ { 2 }$ - 2) + ( $x ^ { 2 }$ - 2) 
C)  ± $\sqrt { 3 }$ 
D)  (1 + e)ln ( $x ^ { 2 }$ - 2) 
A)  (ln + 1)( $x ^ { 2 }$ - 2) 
B)  ln( $x ^ { 2 }$ - 2) + ( $x ^ { 2 }$ - 2) 
C)  ± $\sqrt { 3 }$ 
D)  (1 + e)ln ( $x ^ { 2 }$ - 2)

Question 3

ln( $x ^ { 4 }$ - $x ^ { 3 }$ + 2x +1)Enter your answer as just $\frac { P ( x ) } { Q ( x ) }$ where P and Q are both polynomials in x in standard form.

Accepted Answer

The answer of ln( $x ^ { 4 }$ -...

Question 4

Simplify. 
-( $e ^ { x }$ + $\mathrm { e } ^ { - \mathrm { x } }$ $)^2$ Enter your answer exactly as $e ^ { a }$ + $\mathrm { e } ^ { \mathrm { b } }$ + c (b < a).

Accepted Answer

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

Question 5

$\frac { \ln 3 x } { \ln x }$ Enter your answer as just $\frac { \pm \ln \mathrm { c } } { Q ( \mathrm { x } ) \mathrm { R } ( \ln \mathrm { x } ) }$ where P and R are polynomials in ln x and Q is a polynomial in x, all in standard form.

Accepted Answer

The answer of \(\frac { \ln 3 x } {...

Question 6

f(x) = $\frac { e ^ { x } } { 1 + x ^ { 2 } }$ Enter your answer exactly as $\frac { ( \mathrm { P } ( \mathrm { x } ) ) \mathrm { e } ^ { \mathrm { a } } } { ( \mathrm { Q } ( \mathrm { x } ) ) ^ { \mathrm { n } } }$ where P and Q are polynomials.

Accepted Answer

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

Question 7

$\mathrm { e } ^ { \ln 3 + \ln ( 2 x ) }$ Enter just a standard polynomial in x.

Accepted Answer

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

Question 8

Differentiate. 
-$x ^ { 3 }$ ln x

Accepted Answer

A)  3 $x ^ { 2 }$ ln x + $x ^ { 2 }$ 
B)  ( $3 x ^ { 2 }$ + 1) ln x 
C)  3 $x ^ { 2 }$ ln x 
D)  $x ^ { 2 }$ ln x + $x ^ { 2 }$ 
E)  none of these 
A)  3 $x ^ { 2 }$ ln x + $x ^ { 2 }$ 
B)  ( $3 x ^ { 2 }$ + 1) ln x 
C)  3 $x ^ { 2 }$ ln x 
D)  $x ^ { 2 }$ ln x + $x ^ { 2 }$ 
E)  none of these

Question 9

f(x) = (1 + $x ^ { 2 }$ ) $e ^ { x }$ Enter your answer exactly in the form (P(x)) $e ^ { a }$ where P is a polynomial.

Accepted Answer

The answer of f(x) = (1 + \(x ^ {...

Question 10

Use logarithmic differentiation to differentiate.
-ln $\left( \frac { t + e } { t - e } \right)$ Enter your answer as just $\frac { a e } { t ^ { n } \pm e ^ { m } }$ .

Accepted Answer

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

Question 11

Find the slope of the tangent line to the curve y = $\frac { e ^ { x } } { x }$ at (1, e).
Enter just an integer.

Accepted Answer

The answer of Find the slope of the tangent line...

Question 12

Differentiate. 
-$\sqrt { \ln 3 x }$

Accepted Answer

A)  $\frac { 1 } { 6 x }$ 
B)  $\frac { 1 } { 3 \times \sqrt { \ln 3 x } }$ 
C)  $\frac { 1 } { 2 x \sqrt { \ln 3 x } }$ 
D)  $\frac { 1 } { 6 \times \sqrt { \ln 3 x } }$ 
E)  none of these 
A)  $\frac { 1 } { 6 x }$ 
B)  $\frac { 1 } { 3 \times \sqrt { \ln 3 x } }$ 
C)  $\frac { 1 } { 2 x \sqrt { \ln 3 x } }$ 
D)  $\frac { 1 } { 6 \times \sqrt { \ln 3 x } }$ 
E)  none of these

Question 13

Differentiate. 
-$\frac { e ^ { ( x - 1 ) } } { e ^ { ( x + 1 ) } }$

Accepted Answer

A)  0 
B)  $\frac { 2 \mathrm { e } ^ { \mathrm { x } } } { \left( \mathrm { e } ^ { \mathrm { x } } + 1 \right) ^ { 2 } }$ 
C)  $\frac { e ^ { x } - e ^ { x } \left( e ^ { x } - 1 \right) } { \left( e ^ { x } + 1 \right) }$ 
D)  1 
A)  0 
B)  $\frac { 2 \mathrm { e } ^ { \mathrm { x } } } { \left( \mathrm { e } ^ { \mathrm { x } } + 1 \right) ^ { 2 } }$ 
C)  $\frac { e ^ { x } - e ^ { x } \left( e ^ { x } - 1 \right) } { \left( e ^ { x } + 1 \right) }$ 
D)  1

Question 14

f(x) = x ln(2x - $x ^ { 2 }$ ) at x = 1
Enter your answer as just a real number.

Accepted Answer

The answer of f(x) = x ln(2x - \(x ^...

Question 15

(x + 3ln x $) ^ { 4 }$ at x = 1 Enter just an integer.

Accepted Answer

The answer of (x + 3ln x \() ^ {...

Question 16

$\mathrm { e } ^ { 2 x }$ - $x ^ { 2 }$ Enter your answer exactly as a $\mathrm { e } ^ { \mathrm { b } }$ - c

Accepted Answer

The answer of \(\mathrm { e } ^ { 2...

Question 17

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

Accepted Answer

A)  - $\frac { 2 \left( e ^ { 3 } - 2 x \right) } { 3 }$ 
B)  $e ^ { - 2 x }$ 
C)  $\frac { e ^ { 4 - 2 x } } { 12 }$ 
D)  - $\frac { e ^ { 3 - 2 x } } { 6 }$ 
A)  - $\frac { 2 \left( e ^ { 3 } - 2 x \right) } { 3 }$ 
B)  $e ^ { - 2 x }$ 
C)  $\frac { e ^ { 4 - 2 x } } { 12 }$ 
D)  - $\frac { e ^ { 3 - 2 x } } { 6 }$

Question 18

Simplify
-$2 ^ { x }$ ∙ $3 ^ { x }$ ∙ $5 ^ { x }$ ∙ $7^ x$ Enter your answer exactly as $a^ b$ where a is an integer.

Accepted Answer

The answer of Simplify
-$2 ^ { x }$ ∙ \(3...

Question 19

Simplify
-ln $\frac { x ^ { 5 } y ^ { - 4 } } { w ^ { 3 } z ^ { - 2 } }$

Accepted Answer

A)  5lnx - 4lny - 3lnw - 2lnz 
B)  5lnx + 4lny - 3lnw + 2lnz 
C)  5lnx - 4lny - 3lnw + 2lnz 
D)  5lnx + 4lny - 3lnw - 2lnz 
E)  none of these 
A)  5lnx - 4lny - 3lnw - 2lnz 
B)  5lnx + 4lny - 3lnw + 2lnz 
C)  5lnx - 4lny - 3lnw + 2lnz 
D)  5lnx + 4lny - 3lnw - 2lnz 
E)  none of these

Question 20

Simplify. 
-$\left( e ^ { 2 x } \right) ^ { \frac { 5 } { 3 e ^ { 1 / 2 x } } }$

Accepted Answer

A)  5 $e ^ { 3 / 2 x + 3 }$ 
B)  $5 \mathrm { e } ^ { 3 \mathrm { x } }$ 
C)  $\frac { 5 \mathrm { e } ^ { 2 \mathrm { x } + 3 } } { \mathrm { e } ^ { 1 / 2 \mathrm { x } } }$ 
D)  5 $e ^ { 11 / 2 x }$ 
A)  5 $e ^ { 3 / 2 x + 3 }$ 
B)  $5 \mathrm { e } ^ { 3 \mathrm { x } }$ 
C)  $\frac { 5 \mathrm { e } ^ { 2 \mathrm { x } + 3 } } { \mathrm { e } ^ { 1 / 2 \mathrm { x } } }$ 
D)  5 $e ^ { 11 / 2 x }$

Solve for x. - $e ^ { 4 x ^ { 2 } }$ + $e ^ { ( 2 x ) ^ { 2 } }$ = 6

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

ln( $x ^ { 4 }$ - $x ^ { 3 }$ + 2x +1)Enter your answer as just $\frac { P ( x ) } { Q ( x ) }$ where P and Q are both polynomials in x in standard form.

Simplify. -( $e ^ { x }$ + $\mathrm { e } ^ { - \mathrm { x } }$ $)^2$ Enter your answer exactly as $e ^ { a }$ + $\mathrm { e } ^ { \mathrm { b } }$ + c (b < a).

$\frac { \ln 3 x } { \ln x }$ Enter your answer as just $\frac { \pm \ln \mathrm { c } } { Q ( \mathrm { x } ) \mathrm { R } ( \ln \mathrm { x } ) }$ where P and R are polynomials in ln x and Q is a polynomial in x, all in standard form.

f(x) = $\frac { e ^ { x } } { 1 + x ^ { 2 } }$ Enter your answer exactly as $\frac { ( \mathrm { P } ( \mathrm { x } ) ) \mathrm { e } ^ { \mathrm { a } } } { ( \mathrm { Q } ( \mathrm { x } ) ) ^ { \mathrm { n } } }$ where P and Q are polynomials.

$\mathrm { e } ^ { \ln 3 + \ln ( 2 x ) }$ Enter just a standard polynomial in x.

Differentiate. - $x ^ { 3 }$ ln x

f(x) = (1 + $x ^ { 2 }$ ) $e ^ { x }$ Enter your answer exactly in the form (P(x)) $e ^ { a }$ where P is a polynomial.

Use logarithmic differentiation to differentiate. -ln $\left( \frac { t + e } { t - e } \right)$ Enter your answer as just $\frac { a e } { t ^ { n } \pm e ^ { m } }$ .

Find the slope of the tangent line to the curve y = $\frac { e ^ { x } } { x }$ at (1, e). Enter just an integer.

Differentiate. - $\sqrt { \ln 3 x }$

Differentiate. - $\frac { e ^ { ( x - 1 ) } } { e ^ { ( x + 1 ) } }$

f(x) = x ln(2x - $x ^ { 2 }$ ) at x = 1 Enter your answer as just a real number.

(x + 3ln x $) ^ { 4 }$ at x = 1 Enter just an integer.

$\mathrm { e } ^ { 2 x }$ - $x ^ { 2 }$ Enter your answer exactly as a $\mathrm { e } ^ { \mathrm { b } }$ - c

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

Simplify - $2 ^ { x }$ ∙ $3 ^ { x }$ ∙ $5 ^ { x }$ ∙ $7^ x$ Enter your answer exactly as $a^ b$ where a is an integer.

Simplify -ln $\frac { x ^ { 5 } y ^ { - 4 } } { w ^ { 3 } z ^ { - 2 } }$

Simplify. - $\left( e ^ { 2 x } \right) ^ { \frac { 5 } { 3 e ^ { 1 / 2 x } } }$

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

Solve for x. - e4x2e ^ { 4 x ^ { 2 } }e4x2 + e(2x)2e ^ { ( 2 x ) ^ { 2 } }e(2x)2 = 6

Simplify. -ln( x2x ^ { 2 }x2 - 2) + e ∙ ln( x2x ^ { 2 }x2 - 2)

ln( x4x ^ { 4 }x4 - x3x ^ { 3 }x3 + 2x +1)Enter your answer as just P(x)Q(x)\frac { P ( x ) } { Q ( x ) }Q(x)P(x)​ where P and Q are both polynomials in x in standard form.

Simplify. -( exe ^ { x }ex + e−x\mathrm { e } ^ { - \mathrm { x } }e−x )2)^2)2 Enter your answer exactly as eae ^ { a }ea + eb\mathrm { e } ^ { \mathrm { b } }eb + c (b < a).

ln⁡3xln⁡x\frac { \ln 3 x } { \ln x }lnxln3x​ Enter your answer as just ±ln⁡cQ(x)R(ln⁡x)\frac { \pm \ln \mathrm { c } } { Q ( \mathrm { x } ) \mathrm { R } ( \ln \mathrm { x } ) }Q(x)R(lnx)±lnc​ where P and R are polynomials in ln x and Q is a polynomial in x, all in standard form.

f(x) = ex1+x2\frac { e ^ { x } } { 1 + x ^ { 2 } }1+x2ex​ Enter your answer exactly as (P(x))ea(Q(x))n\frac { ( \mathrm { P } ( \mathrm { x } ) ) \mathrm { e } ^ { \mathrm { a } } } { ( \mathrm { Q } ( \mathrm { x } ) ) ^ { \mathrm { n } } }(Q(x))n(P(x))ea​ where P and Q are polynomials.

eln⁡3+ln⁡(2x)\mathrm { e } ^ { \ln 3 + \ln ( 2 x ) }eln3+ln(2x) Enter just a standard polynomial in x.

Differentiate. - x3x ^ { 3 }x3 ln x

f(x) = (1 + x2x ^ { 2 }x2 ) exe ^ { x }ex Enter your answer exactly in the form (P(x)) eae ^ { a }ea where P is a polynomial.

Use logarithmic differentiation to differentiate. -ln (t+et−e)\left( \frac { t + e } { t - e } \right)(t−et+e​) Enter your answer as just aetn±em\frac { a e } { t ^ { n } \pm e ^ { m } }tn±emae​ .

Find the slope of the tangent line to the curve y = exx\frac { e ^ { x } } { x }xex​ at (1, e). Enter just an integer.

Differentiate. - ln⁡3x\sqrt { \ln 3 x }ln3x​

Differentiate. - e(x−1)e(x+1)\frac { e ^ { ( x - 1 ) } } { e ^ { ( x + 1 ) } }e(x+1)e(x−1)​

f(x) = x ln(2x - x2x ^ { 2 }x2 ) at x = 1 Enter your answer as just a real number.

(x + 3ln x )4) ^ { 4 })4 at x = 1 Enter just an integer.

e2x\mathrm { e } ^ { 2 x }e2x - x2x ^ { 2 }x2 Enter your answer exactly as a eb\mathrm { e } ^ { \mathrm { b } }eb - c

Differentiate. - e(3−2x)3\frac { e ^ { ( 3 - 2 x ) } } { 3 }3e(3−2x)​

Simplify - 2x2 ^ { x }2x ∙ 3x3 ^ { x }3x ∙ 5x5 ^ { x }5x ∙ 7x7^ x7x Enter your answer exactly as aba^ bab where a is an integer.

Simplify -ln x5y−4w3z−2\frac { x ^ { 5 } y ^ { - 4 } } { w ^ { 3 } z ^ { - 2 } }w3z−2x5y−4​

Simplify. - (e2x)53e1/2x\left( e ^ { 2 x } \right) ^ { \frac { 5 } { 3 e ^ { 1 / 2 x } } }(e2x)3e1/2x5​

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

Solve for x. - $e ^ { 4 x ^ { 2 } }$ + $e ^ { ( 2 x ) ^ { 2 } }$ = 6

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

ln( $x ^ { 4 }$ - $x ^ { 3 }$ + 2x +1)Enter your answer as just $\frac { P ( x ) } { Q ( x ) }$ where P and Q are both polynomials in x in standard form.

Simplify. -( $e ^ { x }$ + $\mathrm { e } ^ { - \mathrm { x } }$ $)^2$ Enter your answer exactly as $e ^ { a }$ + $\mathrm { e } ^ { \mathrm { b } }$ + c (b < a).

$\frac { \ln 3 x } { \ln x }$ Enter your answer as just $\frac { \pm \ln \mathrm { c } } { Q ( \mathrm { x } ) \mathrm { R } ( \ln \mathrm { x } ) }$ where P and R are polynomials in ln x and Q is a polynomial in x, all in standard form.

f(x) = $\frac { e ^ { x } } { 1 + x ^ { 2 } }$ Enter your answer exactly as $\frac { ( \mathrm { P } ( \mathrm { x } ) ) \mathrm { e } ^ { \mathrm { a } } } { ( \mathrm { Q } ( \mathrm { x } ) ) ^ { \mathrm { n } } }$ where P and Q are polynomials.

$\mathrm { e } ^ { \ln 3 + \ln ( 2 x ) }$ Enter just a standard polynomial in x.

Differentiate. - $x ^ { 3 }$ ln x

f(x) = (1 + $x ^ { 2 }$ ) $e ^ { x }$ Enter your answer exactly in the form (P(x)) $e ^ { a }$ where P is a polynomial.

Use logarithmic differentiation to differentiate. -ln $\left( \frac { t + e } { t - e } \right)$ Enter your answer as just $\frac { a e } { t ^ { n } \pm e ^ { m } }$ .

Find the slope of the tangent line to the curve y = $\frac { e ^ { x } } { x }$ at (1, e). Enter just an integer.

Differentiate. - $\sqrt { \ln 3 x }$

Differentiate. - $\frac { e ^ { ( x - 1 ) } } { e ^ { ( x + 1 ) } }$

f(x) = x ln(2x - $x ^ { 2 }$ ) at x = 1 Enter your answer as just a real number.

(x + 3ln x $) ^ { 4 }$ at x = 1 Enter just an integer.

$\mathrm { e } ^ { 2 x }$ - $x ^ { 2 }$ Enter your answer exactly as a $\mathrm { e } ^ { \mathrm { b } }$ - c

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

Simplify - $2 ^ { x }$ ∙ $3 ^ { x }$ ∙ $5 ^ { x }$ ∙ $7^ x$ Enter your answer exactly as $a^ b$ where a is an integer.

Simplify -ln $\frac { x ^ { 5 } y ^ { - 4 } } { w ^ { 3 } z ^ { - 2 } }$

Simplify. - $\left( e ^ { 2 x } \right) ^ { \frac { 5 } { 3 e ^ { 1 / 2 x } } }$