Exam 4: Exponential and Logarithmic Functions

The equation of the tangent line to $f ( x ) = e ^ { 7 x } - e ^ { - 7 x }$ at x = 0 is y = 14x.

The function $y = e ^ {5 x }$ is increasing everywhere.

A manufacturer can produce toasters at a cost of $7 apiece and estimates that if they are sold for x dollars apiece, consumers will buy approximately $2,000 e ^ { - 0.2 x }$ toasters per week. At what price should the toasters be sold in order to maximize profit?

Solve for x if $\log _ { 6 } x = 3$ .

The equation of the tangent line to $f ( x ) = 3 e ^ { x }$ at x = 0 is y = 3x + 3.

Solve for x: $5 = 7 - 3 e ^ { - 2 x }$

Use logarithm properties to simplify the expression. $\log _ { 7 } \left[ \frac { x ^ { 7 } ( 8 - 7 x ) ^ { 7 / 2 } } { \sqrt { x ^ { 2 } + x } } \right]$

The equation of the tangent line to f (x) = ln x + 4 at x = 1 is

Determine the monthly car payment for a new car costing $15,624, if there is a down payment of $8,000 and the car is financed over a 8-year period at an annual rate of 9% compounded monthly.

The population density x miles from the center of a city is given by a function of the form $Q ( x ) = A e ^ { - k x }$ . Find this function if it is known that the population density at the center of the city is 9,000 people per square mile and the density 15 miles from the center is 6,000 people per square mile. Round numbers to six decimal places, if necessary.

Solve for x. Round to three decimal places, if necessary. log₃ x = 4

Find all real numbers x that satisfy the given equation. $\left( \frac { 1 } { 9 } \right) ^ { x - 1 } = 3 ^ { 2 + 5 x ^ { 2 } }$

The population density x miles from the center of a certain city is $D ( x ) = 9 e ^ { - 0.06 x }$ thousand people per square mile. What is the population density 5 miles from the center of this city? Round your answer to two decimal places.

Use logarithmic differentiation to find $\frac { d y } { d x }$ where $y = ( 3 x - 1 ) ^ { 8 } ( x + 6 ) ^ { 6 } ( 3 - x ) ^ { 8 }$ .

Use logarithmic differentiation to find $\frac { d y } { d x }$ , where $y = \frac { e ^ { 2 x } ( 2 x - 1 ) ^ { 6 } } { \left( x ^ { 3 } + 5 \right) ^ { 2 } ( 4 - 7 x ) }$ .

Let $f ( x ) = 10 x ^ { 4 } - 160 \ln x$ , for x > 0. Find the minimum value of f for x > 0.

A sum of money, A₀, is invested at a certain fixed interest rate, and this interest is compounded continuously. After 7 years, the money has doubled. The balance at the end of 14 years is $4 A _ { 0 }$ .

A manufacturer can produce radios at a cost of $10 apiece and estimates that if they are sold for x dollars apiece, consumers will buy approximately $200 e ^ { - 0.2 x }$ radios per month. The price at which the manufacturer should sell the radios to maximize the profit is

If an account that earns interest compounded continuously takes $9.5$ years to double in value, how long will it take to triple? Round your answer to one decimal place, if necessary.

The total number of hot dogs sold by a fast-food chain is growing exponentially. If 3 billion have been sold by 1988 and 5 billion by 1990, how many billion will be sold in the year 2000? Round your answer to one decimal place.