Question 1

Find an equation of the tangent line to the graph of y = 2x + ln $\left( \frac { 1 } { x } \right)$ at x = 1.

Accepted Answer

A)  y - 1 = $\left( 2 - \frac { 1 } { x } \right.)$ (x - 1) 
B)  y - 2 = $\left( 2 - \frac { 1 } { x } \right.)$ (x - 1) 
C)  y = 2(x - 1) + 1 
D)  y = x + 1 
A)  y - 1 = $\left( 2 - \frac { 1 } { x } \right.)$ (x - 1) 
B)  y - 2 = $\left( 2 - \frac { 1 } { x } \right.)$ (x - 1) 
C)  y = 2(x - 1) + 1 
D)  y = x + 1

Question 2

Simplify. 
-$\frac { 1 } { e ^ { 5 } }$ ∙ $\frac { \mathrm { e } ^ { 3 } } { 2 }$ ∙ $e ^ { 4 x - 1 }$

Accepted Answer

A)  $\frac { e ^ { 12 x - 3 } } { 2 e ^ { 5 } }$ 
B)  $\frac { 1 } { 2 }$ $e ^ { 4 x - 3 }$ 
C)  $\frac { 1 } { 2 }$ $e ^ { 12 x - 8 }$ 
D)  cannot be simplified 
A)  $\frac { e ^ { 12 x - 3 } } { 2 e ^ { 5 } }$ 
B)  $\frac { 1 } { 2 }$ $e ^ { 4 x - 3 }$ 
C)  $\frac { 1 } { 2 }$ $e ^ { 12 x - 8 }$ 
D)  cannot be simplified

Question 3

Simplify. 
-$e ^ { 1 / 2 }$ ∙ 2 $e ^ { 2 }$ ∙ $5 e ^ { 3 }$

Accepted Answer

A)  10 $\mathrm { e } ^ { 3 }$ 
B)  10 $e ^ { 11 / 2 }$ 
C)  11 $e ^ { 11 / 2 }$ 
D)  cannot be simplified 
A)  10 $\mathrm { e } ^ { 3 }$ 
B)  10 $e ^ { 11 / 2 }$ 
C)  11 $e ^ { 11 / 2 }$ 
D)  cannot be simplified

Question 4

Find the equation of the tangent line to the curve y = $\frac { e ^ { x } } { 1 + e ^ { x } }$ at $\left( 0 , \frac { 1 } { 2 } \right)$ .
Enter your answer in slope-intercept form with any fractions reduced as $\frac { a } { b }$ .

Accepted Answer

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

Question 5

Simplify
-$16 ^ { x }$ $\left( \frac { 1 } { 8 } \right) ^ { x }$ Enter your answer exactly as $2 ^a$ .

Accepted Answer

The answer of Simplify
-$16 ^ { x }$ \(\left( \frac...

Question 6

Simplify
-$\frac { 3 ^ { x } } { 6 ^ { x } }$ ∙ $8 ^ { x }$ ∙ $\left( \frac { 32 } { 4 } \right) ^ { x }$ Enter your answer exactly as $2 ^a$ .

Accepted Answer

The answer of Simplify
-\(\frac { 3 ^ { x }...

Question 7

Simplify. 
-$\frac { \left( e ^ { 1 / 2 x } \right) - 3 / 4 } { e ^ { 2 x - 5 } }$ Enter your answer exactly as $\mathrm { e } ^ { \mathrm { P } } ( \mathrm { x } )$ where P is a polynomial with any fractions in the form $\frac { a } { b }$ in lowest terms.

Accepted Answer

The answer of Simplify. 
-\(\frac { \left( e ^ {...

Question 8

Find the slope of the graph of y = ln(2x + 3 $) ^ { 1 / 2 }$ at the point (3, ln 3).

Accepted Answer

A)  $\frac { \ln 3 } { 2 }$ 
B)  $\frac { 1 } { \ln 3 }$ 
C)  $\frac { 1 } { 2 ( \ln 3 ) + 3 }$ 
D)  $\frac { 1 } { 9 }$ 
A)  $\frac { \ln 3 } { 2 }$ 
B)  $\frac { 1 } { \ln 3 }$ 
C)  $\frac { 1 } { 2 ( \ln 3 ) + 3 }$ 
D)  $\frac { 1 } { 9 }$

Question 9

Simplify
-$3^{10 x}$ ∙ $3^{11 x}$ ∙ $3^{12 x}$ ∙ $3 - 10 x$ Enter your answer exactly as $3^{ a}$ .

Accepted Answer

The answer of Simplify
-$3^{10 x}$ ∙ $3^{11 x}$ ∙ \(3^{12...

Question 10

Choose the one alternative that best completes the statement or answers the question. 
-$\left( \frac { 1 } { 27 } \right) ^ { p } \cdot \left( \frac { 3 } { 27 } \right) ^ { p } \cdot \left( \frac { 1 } { 3 } \right)$

Accepted Answer

A)  $3 ( - 5 p - 1 )$ 
B)  $\left( \frac { 1 } { 3 } \right) ^ { 6 p }$ 
C)  $\frac { 1 } { 3 }$ $\left( \frac { 4 } { 27 } \right) ^ { p }$ 
D)  none of these 
A)  $3 ( - 5 p - 1 )$ 
B)  $\left( \frac { 1 } { 3 } \right) ^ { 6 p }$ 
C)  $\frac { 1 } { 3 }$ $\left( \frac { 4 } { 27 } \right) ^ { p }$ 
D)  none of these

Question 11

Simplify
-$\left( \frac { 1 } { 4 } \right) ^ { 2 x }$ ∙ $\left( \frac { 1 } { 27 } \right) ^ { 3 x }$ ∙ $\left( \frac { 1 } { 64 } \right) ^ { 8 x }$ Enter your answer exactly as $2 ^ { a } \cdot 3 ^ { b }$

Accepted Answer

The answer of Simplify
-\(\left( \frac { 1 } { 4...

Question 12

Simplify
-$y ^ { 4 x }$ ∙ $y ^ { 6 x }$ ∙ $y ^ { x }$ ∙ $y ^ { 4 }$ Enter your answer as $y ^ { P } ( x )$ where P is a polynomial in x in standard form.

Accepted Answer

The answer of Simplify
-$y ^ { 4 x }$ ∙...

Question 13

Simplify
-$7 ^ { - x }$ ∙ $14 ^ { x }$ ∙ $49^{8 x}$ Enter your answer exactly as $2 ^a$ $.7 ^\mathrm {~b}$ .

Accepted Answer

The answer of Simplify
-$7 ^ { - x }$ ∙...

Question 14

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

Accepted Answer

The answer of \(\mathrm { e } ^ { 1...

Question 15

$\mathrm { e } ^ { 3 \mathrm { e } ^ { 2 x } }$ Enter your answer exactly as just a $\mathrm { e } ^ { \mathrm { P } } \left( \mathrm { e } ^ { \mathrm { x } } \right)$ where P is a polynomial in $e ^ { x }$ in standard form.

Accepted Answer

The answer of \(\mathrm { e } ^ { 3...

Find an equation of the tangent line to the graph of y = 2x + ln $\left( \frac { 1 } { x } \right)$ at x = 1.

Simplify. - $\frac { 1 } { e ^ { 5 } }$ ∙ $\frac { \mathrm { e } ^ { 3 } } { 2 }$ ∙ $e ^ { 4 x - 1 }$

Simplify. - $e ^ { 1 / 2 }$ ∙ 2 $e ^ { 2 }$ ∙ $5 e ^ { 3 }$

Find the equation of the tangent line to the curve y = $\frac { e ^ { x } } { 1 + e ^ { x } }$ at $\left( 0 , \frac { 1 } { 2 } \right)$ . Enter your answer in slope-intercept form with any fractions reduced as $\frac { a } { b }$ .

Simplify - $16 ^ { x }$ $\left( \frac { 1 } { 8 } \right) ^ { x }$ Enter your answer exactly as $2 ^a$ .

Simplify - $\frac { 3 ^ { x } } { 6 ^ { x } }$ ∙ $8 ^ { x }$ ∙ $\left( \frac { 32 } { 4 } \right) ^ { x }$ Enter your answer exactly as $2 ^a$ .

Simplify. - $\frac { \left( e ^ { 1 / 2 x } \right) - 3 / 4 } { e ^ { 2 x - 5 } }$ Enter your answer exactly as $\mathrm { e } ^ { \mathrm { P } } ( \mathrm { x } )$ where P is a polynomial with any fractions in the form $\frac { a } { b }$ in lowest terms.

Find the slope of the graph of y = ln(2x + 3 $) ^ { 1 / 2 }$ at the point (3, ln 3).

Simplify - $3^{10 x}$ ∙ $3^{11 x}$ ∙ $3^{12 x}$ ∙ $3 - 10 x$ Enter your answer exactly as $3^{ a}$ .

Choose the one alternative that best completes the statement or answers the question. - $\left( \frac { 1 } { 27 } \right) ^ { p } \cdot \left( \frac { 3 } { 27 } \right) ^ { p } \cdot \left( \frac { 1 } { 3 } \right)$

Simplify - $\left( \frac { 1 } { 4 } \right) ^ { 2 x }$ ∙ $\left( \frac { 1 } { 27 } \right) ^ { 3 x }$ ∙ $\left( \frac { 1 } { 64 } \right) ^ { 8 x }$ Enter your answer exactly as $2 ^ { a } \cdot 3 ^ { b }$

Simplify - $y ^ { 4 x }$ ∙ $y ^ { 6 x }$ ∙ $y ^ { x }$ ∙ $y ^ { 4 }$ Enter your answer as $y ^ { P } ( x )$ where P is a polynomial in x in standard form.

Simplify - $7 ^ { - x }$ ∙ $14 ^ { x }$ ∙ $49^{8 x}$ Enter your answer exactly as $2 ^a$ $.7 ^\mathrm {~b}$ .

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

$\mathrm { e } ^ { 3 \mathrm { e } ^ { 2 x } }$ Enter your answer exactly as just a $\mathrm { e } ^ { \mathrm { P } } \left( \mathrm { e } ^ { \mathrm { x } } \right)$ where P is a polynomial in $e ^ { x }$ in standard form.

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

Find an equation of the tangent line to the graph of y = 2x + ln (1x)\left( \frac { 1 } { x } \right)(x1​) at x = 1.

Simplify. - 1e5\frac { 1 } { e ^ { 5 } }e51​ ∙ e32\frac { \mathrm { e } ^ { 3 } } { 2 }2e3​ ∙ e4x−1e ^ { 4 x - 1 }e4x−1

Simplify. - e1/2e ^ { 1 / 2 }e1/2 ∙ 2 e2e ^ { 2 }e2 ∙ 5e35 e ^ { 3 }5e3

Find the equation of the tangent line to the curve y = ex1+ex\frac { e ^ { x } } { 1 + e ^ { x } }1+exex​ at (0,12)\left( 0 , \frac { 1 } { 2 } \right)(0,21​) . Enter your answer in slope-intercept form with any fractions reduced as ab\frac { a } { b }ba​ .

Simplify - 16x16 ^ { x }16x (18)x\left( \frac { 1 } { 8 } \right) ^ { x }(81​)x Enter your answer exactly as 2a2 ^a2a .

Simplify - 3x6x\frac { 3 ^ { x } } { 6 ^ { x } }6x3x​ ∙ 8x8 ^ { x }8x ∙ (324)x\left( \frac { 32 } { 4 } \right) ^ { x }(432​)x Enter your answer exactly as 2a2 ^a2a .

Find the slope of the graph of y = ln(2x + 3 )1/2) ^ { 1 / 2 })1/2 at the point (3, ln 3).

Simplify - 310x3^{10 x}310x ∙ 311x3^{11 x}311x ∙ 312x3^{12 x}312x ∙ 3−10x3 - 10 x3−10x Enter your answer exactly as 3a3^{ a}3a .

Choose the one alternative that best completes the statement or answers the question. - (127)p⋅(327)p⋅(13)\left( \frac { 1 } { 27 } \right) ^ { p } \cdot \left( \frac { 3 } { 27 } \right) ^ { p } \cdot \left( \frac { 1 } { 3 } \right)(271​)p⋅(273​)p⋅(31​)

Simplify - (14)2x\left( \frac { 1 } { 4 } \right) ^ { 2 x }(41​)2x ∙ (127)3x\left( \frac { 1 } { 27 } \right) ^ { 3 x }(271​)3x ∙ (164)8x\left( \frac { 1 } { 64 } \right) ^ { 8 x }(641​)8x Enter your answer exactly as 2a⋅3b2 ^ { a } \cdot 3 ^ { b }2a⋅3b

Simplify - y4xy ^ { 4 x }y4x ∙ y6xy ^ { 6 x }y6x ∙ yxy ^ { x }yx ∙ y4y ^ { 4 }y4 Enter your answer as yP(x)y ^ { P } ( x )yP(x) where P is a polynomial in x in standard form.

Simplify - 7−x7 ^ { - x }7−x ∙ 14x14 ^ { x }14x ∙ 498x49^{8 x}498x Enter your answer exactly as 2a2 ^a2a .7 b.7 ^\mathrm {~b}.7 b .

e1+x=2e2x\mathrm { e } ^ { 1 + x } = 2 ^ { \mathrm { e } ^ { 2 x } }e1+x=2e2x Enter your answer exactly as just a ± ln b (a, b integers).

e3e2x\mathrm { e } ^ { 3 \mathrm { e } ^ { 2 x } }e3e2x Enter your answer exactly as just a eP(ex)\mathrm { e } ^ { \mathrm { P } } \left( \mathrm { e } ^ { \mathrm { x } } \right)eP(ex) where P is a polynomial in exe ^ { x }ex in standard form.

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

Find an equation of the tangent line to the graph of y = 2x + ln $\left( \frac { 1 } { x } \right)$ at x = 1.

Simplify. - $\frac { 1 } { e ^ { 5 } }$ ∙ $\frac { \mathrm { e } ^ { 3 } } { 2 }$ ∙ $e ^ { 4 x - 1 }$

Simplify. - $e ^ { 1 / 2 }$ ∙ 2 $e ^ { 2 }$ ∙ $5 e ^ { 3 }$

Find the equation of the tangent line to the curve y = $\frac { e ^ { x } } { 1 + e ^ { x } }$ at $\left( 0 , \frac { 1 } { 2 } \right)$ . Enter your answer in slope-intercept form with any fractions reduced as $\frac { a } { b }$ .

Simplify - $16 ^ { x }$ $\left( \frac { 1 } { 8 } \right) ^ { x }$ Enter your answer exactly as $2 ^a$ .

Simplify - $\frac { 3 ^ { x } } { 6 ^ { x } }$ ∙ $8 ^ { x }$ ∙ $\left( \frac { 32 } { 4 } \right) ^ { x }$ Enter your answer exactly as $2 ^a$ .

Find the slope of the graph of y = ln(2x + 3 $) ^ { 1 / 2 }$ at the point (3, ln 3).

Simplify - $3^{10 x}$ ∙ $3^{11 x}$ ∙ $3^{12 x}$ ∙ $3 - 10 x$ Enter your answer exactly as $3^{ a}$ .

Choose the one alternative that best completes the statement or answers the question. - $\left( \frac { 1 } { 27 } \right) ^ { p } \cdot \left( \frac { 3 } { 27 } \right) ^ { p } \cdot \left( \frac { 1 } { 3 } \right)$

Simplify - $\left( \frac { 1 } { 4 } \right) ^ { 2 x }$ ∙ $\left( \frac { 1 } { 27 } \right) ^ { 3 x }$ ∙ $\left( \frac { 1 } { 64 } \right) ^ { 8 x }$ Enter your answer exactly as $2 ^ { a } \cdot 3 ^ { b }$

Simplify - $y ^ { 4 x }$ ∙ $y ^ { 6 x }$ ∙ $y ^ { x }$ ∙ $y ^ { 4 }$ Enter your answer as $y ^ { P } ( x )$ where P is a polynomial in x in standard form.

Simplify - $7 ^ { - x }$ ∙ $14 ^ { x }$ ∙ $49^{8 x}$ Enter your answer exactly as $2 ^a$ $.7 ^\mathrm {~b}$ .

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

$\mathrm { e } ^ { 3 \mathrm { e } ^ { 2 x } }$ Enter your answer exactly as just a $\mathrm { e } ^ { \mathrm { P } } \left( \mathrm { e } ^ { \mathrm { x } } \right)$ where P is a polynomial in $e ^ { x }$ in standard form.