Question 1

Find the derivative of the following function. $y = 5 e ^ { 3 x ^ { 2 } - 3 }$

Accepted Answer

A)  $y ^ { \prime } = 15 x e ^ { 3 x ^ { 2 } - 3 }$ 
B)  $y ^ { \prime } = 30 x e ^ { 3 x ^ { 2 } - 3 }$ 
C)  $y ^ { \prime } = 30 e ^ { 3 x ^ { 2 } - 3 }$ 
D)  $y ^ { \prime } = 15 e ^ { 3 x ^ { 2 } - 3 }$ 
E)  $y ^ { \prime } = 10 x e ^ { 3 x ^ { 2 } - 3 }$ 
A)  $y ^ { \prime } = 15 x e ^ { 3 x ^ { 2 } - 3 }$ 
B)  $y ^ { \prime } = 30 x e ^ { 3 x ^ { 2 } - 3 }$ 
C)  $y ^ { \prime } = 30 e ^ { 3 x ^ { 2 } - 3 }$ 
D)  $y ^ { \prime } = 15 e ^ { 3 x ^ { 2 } - 3 }$ 
E)  $y ^ { \prime } = 10 x e ^ { 3 x ^ { 2 } - 3 }$ B

Question 2

Find the derivative of the following function. $y = 3 e ^ { 6 \sqrt { x } } + 6$

Accepted Answer

A)  $y ^ { \prime } = \frac { 9 e ^ { 6 \sqrt { x } } } { \sqrt { x } }$ 
B)  $y ^ { \prime } = \frac { 3 e ^ { 6 \sqrt { x } } } { \sqrt { x } }$ 
C)  $y ^ { \prime } = \frac { 18 e ^ { 6 \sqrt { x } } } { \sqrt { x } }$ 
D)  $y ^ { \prime } = \frac { 3 e ^ { 6 \sqrt { x } } } { \sqrt { x } }$ 
E)  $y ^ { \prime } = \frac { 6 e ^ { 6 \sqrt { x } } } { \sqrt { x } }$ 
A)  $y ^ { \prime } = \frac { 9 e ^ { 6 \sqrt { x } } } { \sqrt { x } }$ 
B)  $y ^ { \prime } = \frac { 3 e ^ { 6 \sqrt { x } } } { \sqrt { x } }$ 
C)  $y ^ { \prime } = \frac { 18 e ^ { 6 \sqrt { x } } } { \sqrt { x } }$ 
D)  $y ^ { \prime } = \frac { 3 e ^ { 6 \sqrt { x } } } { \sqrt { x } }$ 
E)  $y ^ { \prime } = \frac { 6 e ^ { 6 \sqrt { x } } } { \sqrt { x } }$ A

Question 3

After t years, the remaining mass y(in grams) of 16 grams of a radioactive element whose half-life is 32 years is given by $y = 16 \left( \frac { 1 } { 2 } \right) ^ { t / 32 }$ , for $t \geq 0$ . How much of the initial mass remains after 96 years? Round your answer to two decimal places.

Accepted Answer

A) 4.00 grams 
B) 3.20 grams 
C) 4.30 grams 
D) 4.90 grams 
E) 2.00 grams 
A) 4.00 grams 
B) 3.20 grams 
C) 4.30 grams 
D) 4.90 grams 
E) 2.00 grams E

Question 4

Solve the following equation for $x$ accurate to three decimal places. $\ln x ^ { - 2 } = 3$

Accepted Answer

A)  $x = - 1.500$ 
B)  $x = 0.223$ 
C)  $x = 1.405$ 
D)  $x = 0.513$ 
E)  $x = 4.621$ 
A)  $x = - 1.500$ 
B)  $x = 0.223$ 
C)  $x = 1.405$ 
D)  $x = 0.513$ 
E)  $x = 4.621$

Question 5

Find $y ^ { \prime }$ . $y = 4 ( \ln x ) ^ { - 9 }$

Accepted Answer

A)  $- \frac { 16 } { x ( \ln x ) ^ { 10 } }$ 
B)  $- \frac { 81 } { x ( \ln x ) ^ { 10 } }$ 
C)  $- \frac { 36 } { x ( \ln x ) ^ { 10 } }$ 
D)  $- \frac { 36 } { x ( \ln x ) ^ { 8 } }$ 
E)  $- \frac { 81 } { x ( \ln x ) ^ { 8 } }$ 
A)  $- \frac { 16 } { x ( \ln x ) ^ { 10 } }$ 
B)  $- \frac { 81 } { x ( \ln x ) ^ { 10 } }$ 
C)  $- \frac { 36 } { x ( \ln x ) ^ { 10 } }$ 
D)  $- \frac { 36 } { x ( \ln x ) ^ { 8 } }$ 
E)  $- \frac { 81 } { x ( \ln x ) ^ { 8 } }$

Question 6

Find the derivative of the function $y = \ln \sqrt { x ^ { 2 } - 7 }$ .

Accepted Answer

A)  $\frac { d y } { d x } = \frac { 1 } { 2 x - 7 }$ 
B)  $\frac { d y } { d x } = \frac { x } { 2 x - 7 }$ 
C)  $\frac { d y } { d x } = \frac { 1 } { x ^ { 2 } - 7 }$ 
D)  $\frac { d y } { d x } = \frac { x } { x ^ { 2 } - 7 }$ 
E)  $\frac { d y } { d x } = \frac { 2 x } { \sqrt { x ^ { 2 } - 7 } }$ 
A)  $\frac { d y } { d x } = \frac { 1 } { 2 x - 7 }$ 
B)  $\frac { d y } { d x } = \frac { x } { 2 x - 7 }$ 
C)  $\frac { d y } { d x } = \frac { 1 } { x ^ { 2 } - 7 }$ 
D)  $\frac { d y } { d x } = \frac { x } { x ^ { 2 } - 7 }$ 
E)  $\frac { d y } { d x } = \frac { 2 x } { \sqrt { x ^ { 2 } - 7 } }$

Question 7

Write the logarithmic equation $\ln$ $1.3 = 0.2624 \mathrm {~K}$ as an exponential equation.

Accepted Answer

A)  $e ^ { - 0.2624 \mathrm {~K} } = 1.3$ 
B)  $e ^ { - 1.3 } = 0.2624 \mathrm {~K}$ 
C)  $e ^ { 0.2624 \mathrm {~K} } = 1.3$ 
D)  $e ^ { 1.3 } = 0.2624 \mathrm {~K}$ 
E)  $e ^ { 0.2624 \mathrm {~K} } = - 1.3$ 
A)  $e ^ { - 0.2624 \mathrm {~K} } = 1.3$ 
B)  $e ^ { - 1.3 } = 0.2624 \mathrm {~K}$ 
C)  $e ^ { 0.2624 \mathrm {~K} } = 1.3$ 
D)  $e ^ { 1.3 } = 0.2624 \mathrm {~K}$ 
E)  $e ^ { 0.2624 \mathrm {~K} } = - 1.3$

Question 8

Future value. The future value that accrues when $900 is invested at 5%, compounded continuously, is $s ( t ) = 900 e ^ { 005 t }$ , where t is the number of years. At what rate is the money in this account growing when $t = 9 ?$

Accepted Answer

A) $14.11 per year 
B) $49.24 per year 
C) $1411.48 per year 
D) $941.43 per year 
E) $70.57 per year 
A) $14.11 per year 
B) $49.24 per year 
C) $1411.48 per year 
D) $941.43 per year 
E) $70.57 per year

Question 9

Find the derivative of the following function. $y = \ln 5 x$

Accepted Answer

A)  $\frac { 1 } { x }$ 
B)  $\frac { 5 } { x }$ 
C)  $\frac { 1 } { 5 x }$ 
D)  $\frac { 1 } { x ^ { 2 } }$ 
E)  $\frac { 1 } { 5 x ^ { 2 } }$ 
A)  $\frac { 1 } { x }$ 
B)  $\frac { 5 } { x }$ 
C)  $\frac { 1 } { 5 x }$ 
D)  $\frac { 1 } { x ^ { 2 } }$ 
E)  $\frac { 1 } { 5 x ^ { 2 } }$

Question 10

Simplify $\ln e ^ { - 6 x ^ { 4 } }$

Accepted Answer

A)  $- 6 x ^ { 4 }$ 
B)  $- 24 x$ 
C)  $- 24 x ^ { 4 }$ 
D)  $- 6 x$ 
E)  $x ^ { 4 }$ 
A)  $- 6 x ^ { 4 }$ 
B)  $- 24 x$ 
C)  $- 24 x ^ { 4 }$ 
D)  $- 6 x$ 
E)  $x ^ { 4 }$

Question 11

Find $\frac { d y } { d x } \text { if } y = \ln \left( x ^ { 6 } \left( x ^ { 2 } - x + 4 \right) \right)$ .

Accepted Answer

A)  $\frac { 8 x ^ { 2 } - 7 x + 24 } { x \left( x ^ { 2 } - x + 4 \right) }$ 
B)  $\frac { 8 x ^ { 2 } + 7 x - 24 } { x \left( x ^ { 2 } - x + 4 \right) }$ 
C)  $\frac { 8 x ^ { 2 } + 7 x + 6 } { x \left( x ^ { 2 } - x + 4 \right) }$ 
D)  $\frac { 2 x ^ { 2 } - 7 x - 6 } { x \left( x ^ { 2 } - x + 4 \right) }$ 
E)  $\frac { 2 x ^ { 2 } + 7 x + 24 } { x \left( x ^ { 2 } - x + 4 \right) }$ 
A)  $\frac { 8 x ^ { 2 } - 7 x + 24 } { x \left( x ^ { 2 } - x + 4 \right) }$ 
B)  $\frac { 8 x ^ { 2 } + 7 x - 24 } { x \left( x ^ { 2 } - x + 4 \right) }$ 
C)  $\frac { 8 x ^ { 2 } + 7 x + 6 } { x \left( x ^ { 2 } - x + 4 \right) }$ 
D)  $\frac { 2 x ^ { 2 } - 7 x - 6 } { x \left( x ^ { 2 } - x + 4 \right) }$ 
E)  $\frac { 2 x ^ { 2 } + 7 x + 24 } { x \left( x ^ { 2 } - x + 4 \right) }$

Question 12

Find the derivative of the following function. $y = 8 ^ { 9 x + 1 }$

Accepted Answer

A)  $y ^ { \prime } = ( 8 \ln 9 ) 8 ^ { 9 x + 1 }$ 
B)  $y ^ { \prime } = ( 9 \ln 8 ) 8 ^ { 9 x + 1 }$ 
C)  $ y ^ { \prime } = ( \ln 9 ) ^ { 9 x + 1 }$ 
D)  $y ^ { \prime } = ( \ln 8 ) 8 ^ { 9 x + 1 }$ 
E)  $y ^ { \prime } = ( \ln 72 ) 8 ^ { 9 x + 1 }$ 
A)  $y ^ { \prime } = ( 8 \ln 9 ) 8 ^ { 9 x + 1 }$ 
B)  $y ^ { \prime } = ( 9 \ln 8 ) 8 ^ { 9 x + 1 }$ 
C)  $ y ^ { \prime } = ( \ln 9 ) ^ { 9 x + 1 }$ 
D)  $y ^ { \prime } = ( \ln 8 ) 8 ^ { 9 x + 1 }$ 
E)  $y ^ { \prime } = ( \ln 72 ) 8 ^ { 9 x + 1 }$

Question 13

Write the following expression as a logarithm of a single quantity. $\ln x - 16 \ln \left( x ^ { 2 } + 1 \right)$

Accepted Answer

A)  $\ln \left( \frac { - 16 x } { \left( x ^ { 2 } + 1 \right) } \right)$ 
B)  $\ln \left( \frac { x } { 16 \left( x ^ { 2 } + 1 \right) } \right)$ 
C)  $\ln \left( x - 16 \left( x ^ { 2 } + 1 \right) \right)$ 
D)  $\ln \left( \frac { x } { \left( x ^ { 2 } + 1 \right) ^ { 16 } } \right)$ 
E)  $\ln \left( \frac { - 16 x } { x ^ { 2 } + 1 } \right)$ 
A)  $\ln \left( \frac { - 16 x } { \left( x ^ { 2 } + 1 \right) } \right)$ 
B)  $\ln \left( \frac { x } { 16 \left( x ^ { 2 } + 1 \right) } \right)$ 
C)  $\ln \left( x - 16 \left( x ^ { 2 } + 1 \right) \right)$ 
D)  $\ln \left( \frac { x } { \left( x ^ { 2 } + 1 \right) ^ { 16 } } \right)$ 
E)  $\ln \left( \frac { - 16 x } { x ^ { 2 } + 1 } \right)$

Question 14

Find an equation of the tangent line to the graph of $y = e ^ { 5 x }$ at the point (0,1) .

Accepted Answer

A)  $y = x + 1$ 
B)  $y = \ln ( 5 ) x + 1$ 
C)  $y = 6 x + 1$ 
D)  $y = 5 x + 1$ 
E)  $y = 5 x - 1$ 
A)  $y = x + 1$ 
B)  $y = \ln ( 5 ) x + 1$ 
C)  $y = 6 x + 1$ 
D)  $y = 5 x + 1$ 
E)  $y = 5 x - 1$

Question 15

With an annual rate of inflation of 4% over the next 10 years, the approximate cost of goods or services during any year in the decade is given by $C ( t ) = P ( 1.04 ) ^ { t } , 0 \leq t \leq 10$ where is the time (in years) and is the present cost. The price of an oil change for a car is presently $24.95.Estimate the price 10 years from now.

Accepted Answer

A) $37.09 
B) $36.93 
C) $89.00 
D) $63.90 
A) $37.09 
B) $36.93 
C) $89.00 
D) $63.90

Question 16

Sketch the graph of the function $f ( x ) = \ln | x |$ .

Accepted Answer

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

Question 17

If $x - 9 x e ^ { 4 y } = 5 , \text { find } d y / d x$

Accepted Answer

A)  $\frac { d y } { d x } = \frac { 1 - 9 e ^ { 4 y } } { 72 e ^ { 4 y } }$ 
B)  $\frac { d y } { d x } = \frac { 1 + 4 e ^ { 4 y } } { 36 x e ^ { 4 y } }$ 
C)  $\frac { d y } { d x } = \frac { 1 - 9 e ^ { 4 y } } { 36 x e ^ { 4 y } }$ 
D)  $\frac { d y } { d x } = \frac { 1 + 9 e ^ { 4 y } } { 72 x e ^ { 4 y } }$ 
E)  $\frac { d y } { d x } = \frac { 1 - 4 e ^ { 4 y } } { 36 e ^ { 4 y } }$ 
A)  $\frac { d y } { d x } = \frac { 1 - 9 e ^ { 4 y } } { 72 e ^ { 4 y } }$ 
B)  $\frac { d y } { d x } = \frac { 1 + 4 e ^ { 4 y } } { 36 x e ^ { 4 y } }$ 
C)  $\frac { d y } { d x } = \frac { 1 - 9 e ^ { 4 y } } { 36 x e ^ { 4 y } }$ 
D)  $\frac { d y } { d x } = \frac { 1 + 9 e ^ { 4 y } } { 72 x e ^ { 4 y } }$ 
E)  $\frac { d y } { d x } = \frac { 1 - 4 e ^ { 4 y } } { 36 e ^ { 4 y } }$

Question 18

After t years, the value of a car that originally cost $\$$ 17,000 depreciates so that each year it is worth $\frac { 3 } { 4 }$ of its value for the previous year. Find a model for V(t), the value of the car after t years.

Accepted Answer

A) V(t)= 17,000 $\left( \frac { 3 ^ { t } } { 4 } \right)$ 
B) V(t)= 17,000t $\left( \frac { 3 } { 4 } \right)$ 
C) V(t)= 17,000 $\left( \frac { 3 } { 4 } \right) ^ { t }$ 
D) V(t)= 17,000 $\left( \frac { 3 } { 4 ^ { t } } \right)$ 
E) V(t)= 17,000t $\left( \frac { 3 } { 4 } \right) ^ { t }$ 
A) V(t)= 17,000 $\left( \frac { 3 ^ { t } } { 4 } \right)$ 
B) V(t)= 17,000t $\left( \frac { 3 } { 4 } \right)$ 
C) V(t)= 17,000 $\left( \frac { 3 } { 4 } \right) ^ { t }$ 
D) V(t)= 17,000 $\left( \frac { 3 } { 4 ^ { t } } \right)$ 
E) V(t)= 17,000t $\left( \frac { 3 } { 4 } \right) ^ { t }$

Question 19

Use the given information to write an exponential equation for y. Does the function represent exponential growth or exponential decay? $\frac { d y } { d t } = 2 y , y = 10 \text { when } t = 0$

Accepted Answer

A)  $y = 10 e ^ { 2 t } ,  \text {expontial decay }$ 
B)  $y = 10 e ^ { 2 t } ,  \text {expontial growth }$ 
C)  $y = 10 e ^ { t } , \text { expontial decay }$ 
D)  $y = e ^ { 2 t } , \text { expontial growth }$ 
A)  $y = 10 e ^ { 2 t } ,  \text {expontial decay }$ 
B)  $y = 10 e ^ { 2 t } ,  \text {expontial growth }$ 
C)  $y = 10 e ^ { t } , \text { expontial decay }$ 
D)  $y = e ^ { 2 t } , \text { expontial growth }$

Question 20

Write the expression $5 \ln ( x ) + 2 \ln ( x + 3 ) - 5 \ln ( x - 3 )$ as the logarithm of a single quantity.

Accepted Answer

A)  $\ln ( 2 x + 21 )$ 
B)  $\ln \left[ \frac { 10 \left( x ^ { 2 } + x \right) } { 5 ( x - 3 ) } \right]$ 
C)  $\ln \left( x ^ { 5 } ( x + 3 ) ^ { 2 } ( x - 3 ) ^ { 5 } \right)$ 
D)  $\ln \left[ \frac { x ^ { 5 } ( x + 3 ) ^ { 2 } } { ( x - 3 ) ^ { 5 } } \right]$ 
E)  $\ln \left[ \frac { x ^ { 5 } } { ( x + 3 ) ^ { 2 } ( x - 3 ) ^ { 5 } } \right]$ 
A)  $\ln ( 2 x + 21 )$ 
B)  $\ln \left[ \frac { 10 \left( x ^ { 2 } + x \right) } { 5 ( x - 3 ) } \right]$ 
C)  $\ln \left( x ^ { 5 } ( x + 3 ) ^ { 2 } ( x - 3 ) ^ { 5 } \right)$ 
D)  $\ln \left[ \frac { x ^ { 5 } ( x + 3 ) ^ { 2 } } { ( x - 3 ) ^ { 5 } } \right]$ 
E)  $\ln \left[ \frac { x ^ { 5 } } { ( x + 3 ) ^ { 2 } ( x - 3 ) ^ { 5 } } \right]$

Find the derivative of the following function. $y = 5 e ^ { 3 x ^ { 2 } - 3 }$

Find the derivative of the following function. $y = 3 e ^ { 6 \sqrt { x } } + 6$

After t years, the remaining mass y(in grams) of 16 grams of a radioactive element whose half-life is 32 years is given by $y = 16 \left( \frac { 1 } { 2 } \right) ^ { t / 32 }$ , for $t \geq 0$ . How much of the initial mass remains after 96 years? Round your answer to two decimal places.

Solve the following equation for $x$ accurate to three decimal places. $\ln x ^ { - 2 } = 3$

Find $y ^ { \prime }$ . $y = 4 ( \ln x ) ^ { - 9 }$

Find the derivative of the function $y = \ln \sqrt { x ^ { 2 } - 7 }$ .

Write the logarithmic equation $\ln$ $1.3 = 0.2624 \mathrm {~K}$ as an exponential equation.

Future value. The future value that accrues when $900 is invested at 5%, compounded continuously, is $s ( t ) = 900 e ^ { 005 t }$ , where t is the number of years. At what rate is the money in this account growing when $t = 9 ?$

Find the derivative of the following function. $y = \ln 5 x$

Simplify $\ln e ^ { - 6 x ^ { 4 } }$

Find $\frac { d y } { d x } \text { if } y = \ln \left( x ^ { 6 } \left( x ^ { 2 } - x + 4 \right) \right)$ .

Find the derivative of the following function. $y = 8 ^ { 9 x + 1 }$

Write the following expression as a logarithm of a single quantity. $\ln x - 16 \ln \left( x ^ { 2 } + 1 \right)$

Find an equation of the tangent line to the graph of $y = e ^ { 5 x }$ at the point (0,1) .

Sketch the graph of the function $f ( x ) = \ln | x |$ .

If $x - 9 x e ^ { 4 y } = 5 , \text { find } d y / d x$

After t years, the value of a car that originally cost $\$$ 17,000 depreciates so that each year it is worth $\frac { 3 } { 4 }$ of its value for the previous year. Find a model for V(t), the value of the car after t years.

Use the given information to write an exponential equation for y. Does the function represent exponential growth or exponential decay? $\frac { d y } { d t } = 2 y , y = 10 \text { when } t = 0$

Write the expression $5 \ln ( x ) + 2 \ln ( x + 3 ) - 5 \ln ( x - 3 )$ as the logarithm of a single quantity.

Fundamental Concepts of Algebra

Equations and Inequalities

Functions and Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Systems of Equations and Inequalities

Matrices and Determinants

Limits and Derivatives

Applications of the Derivative

Further Applications of the Derivative

Integration and Its Applications

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Exam 11: Derivatives of Exponential and Logarithmic Functions

Find the derivative of the following function. y=5e3x2−3y = 5 e ^ { 3 x ^ { 2 } - 3 }y=5e3x2−3

Find the derivative of the following function. y=3e6x+6y = 3 e ^ { 6 \sqrt { x } } + 6y=3e6x​+6

Solve the following equation for xxx accurate to three decimal places. ln⁡x−2=3\ln x ^ { - 2 } = 3lnx−2=3

Find y′y ^ { \prime }y′ . y=4(ln⁡x)−9y = 4 ( \ln x ) ^ { - 9 }y=4(lnx)−9

Find the derivative of the function y=ln⁡x2−7y = \ln \sqrt { x ^ { 2 } - 7 }y=lnx2−7​ .

Write the logarithmic equation ln⁡\lnln 1.3=0.2624 K1.3 = 0.2624 \mathrm {~K}1.3=0.2624 K as an exponential equation.

Future value. The future value that accrues when $900 is invested at 5%, compounded continuously, is s(t)=900e005ts ( t ) = 900 e ^ { 005 t }s(t)=900e005t , where t is the number of years. At what rate is the money in this account growing when t=9?t = 9 ?t=9?

Find the derivative of the following function. y=ln⁡5xy = \ln 5 xy=ln5x

Simplify ln⁡e−6x4\ln e ^ { - 6 x ^ { 4 } }lne−6x4

Find dydx if y=ln⁡(x6(x2−x+4))\frac { d y } { d x } \text { if } y = \ln \left( x ^ { 6 } \left( x ^ { 2 } - x + 4 \right) \right)dxdy​ if y=ln(x6(x2−x+4)) .

Find the derivative of the following function. y=89x+1y = 8 ^ { 9 x + 1 }y=89x+1

Write the following expression as a logarithm of a single quantity. ln⁡x−16ln⁡(x2+1)\ln x - 16 \ln \left( x ^ { 2 } + 1 \right)lnx−16ln(x2+1)

Find an equation of the tangent line to the graph of y=e5xy = e ^ { 5 x }y=e5x at the point (0,1) .

Sketch the graph of the function f(x)=ln⁡∣x∣f ( x ) = \ln | x |f(x)=ln∣x∣ .

If x−9xe4y=5, find dy/dxx - 9 x e ^ { 4 y } = 5 , \text { find } d y / d xx−9xe4y=5, find dy/dx

After t years, the value of a car that originally cost $\$$ 17,000 depreciates so that each year it is worth 34\frac { 3 } { 4 }43​ of its value for the previous year. Find a model for V(t), the value of the car after t years.

Use the given information to write an exponential equation for y. Does the function represent exponential growth or exponential decay? dydt=2y,y=10 when t=0\frac { d y } { d t } = 2 y , y = 10 \text { when } t = 0dtdy​=2y,y=10 when t=0

Write the expression 5ln⁡(x)+2ln⁡(x+3)−5ln⁡(x−3)5 \ln ( x ) + 2 \ln ( x + 3 ) - 5 \ln ( x - 3 )5ln(x)+2ln(x+3)−5ln(x−3) as the logarithm of a single quantity.

Fundamental Concepts of Algebra

Equations and Inequalities

Functions and Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Systems of Equations and Inequalities

Matrices and Determinants

Limits and Derivatives

Applications of the Derivative

Further Applications of the Derivative

Integration and Its Applications

Techniques of Integration

Functions of Several Variables

Trigonometric Functions Web

Series and Taylor Polynomials Web

Probability Web

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Find the derivative of the following function. $y = 5 e ^ { 3 x ^ { 2 } - 3 }$

Find the derivative of the following function. $y = 3 e ^ { 6 \sqrt { x } } + 6$

Solve the following equation for $x$ accurate to three decimal places. $\ln x ^ { - 2 } = 3$

Find $y ^ { \prime }$ . $y = 4 ( \ln x ) ^ { - 9 }$

Find the derivative of the function $y = \ln \sqrt { x ^ { 2 } - 7 }$ .

Write the logarithmic equation $\ln$ $1.3 = 0.2624 \mathrm {~K}$ as an exponential equation.

Future value. The future value that accrues when $900 is invested at 5%, compounded continuously, is $s ( t ) = 900 e ^ { 005 t }$ , where t is the number of years. At what rate is the money in this account growing when $t = 9 ?$

Find the derivative of the following function. $y = \ln 5 x$

Simplify $\ln e ^ { - 6 x ^ { 4 } }$

Find $\frac { d y } { d x } \text { if } y = \ln \left( x ^ { 6 } \left( x ^ { 2 } - x + 4 \right) \right)$ .

Find the derivative of the following function. $y = 8 ^ { 9 x + 1 }$

Write the following expression as a logarithm of a single quantity. $\ln x - 16 \ln \left( x ^ { 2 } + 1 \right)$

Find an equation of the tangent line to the graph of $y = e ^ { 5 x }$ at the point (0,1) .

Sketch the graph of the function $f ( x ) = \ln | x |$ .

If $x - 9 x e ^ { 4 y } = 5 , \text { find } d y / d x$

After t years, the value of a car that originally cost $\$$ 17,000 depreciates so that each year it is worth $\frac { 3 } { 4 }$ of its value for the previous year. Find a model for V(t), the value of the car after t years.

Use the given information to write an exponential equation for y. Does the function represent exponential growth or exponential decay? $\frac { d y } { d t } = 2 y , y = 10 \text { when } t = 0$

Write the expression $5 \ln ( x ) + 2 \ln ( x + 3 ) - 5 \ln ( x - 3 )$ as the logarithm of a single quantity.