Question 1

A landlord rents an apartment building with 250 apartments. The monthly profit, in dollars, can be modeled by  , where x is the number of apartments rented. How many apartments should be rented to maximize profit?

Accepted Answer

The answer of A landlord rents an apartment building with...

Question 2

Find the derivative of  .
A) 
B) 
C) 
D)

Accepted Answer

The answer of Find the derivative of  .
A) 
B)...

Question 3

Find the equation of the tangent line to the curve  which passes through the origin.

Accepted Answer

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

Question 4

Consider the function  . We know that  is concave down when _____ < x < _____.

Accepted Answer

The answer of Consider the function  . We know...

Question 5

Find the derivative of  .
A) 
B) 
C) 
D)

Accepted Answer

Without knowing what the variable is, we can't give a specific answer for the derivative, but we can see that the last 9 characters of the original expression are constant and therefore don't affect the derivative. The first part, 11eb69ff_fe50_147a_af94_774f9ca0e0fe, represents the variable, which we'll call x. The derivative of x with respect to x (i.e. the derivative of x) is 1. Therefore, the derivative of the given expression with respect to x is 1, followed by the constant 11, followed by the constant 9. Both options A and C change the second-to-last character, which would change the value of the constant. Option D changes the fourth-to-last character, which also affects the value of the constant. Only option B changes the first character after the constant section, which is where the derivative would be affected.

Question 6

The population of a town is approximated by the function  , where t is the number of years since 1980. Find  . Round to the nearest whole number.

Accepted Answer

The answer of The population of a town is approximated...

Question 7

Find the derivative of 
A) 
B) 
C) 
D)

Accepted Answer

The answer of Find the derivative of 
A) 
B) 
C)...

Question 8

Given  and  , find  .

Accepted Answer

The answer of Given  and  , find ...

Question 9

Consider the function  . We know that  is increasing when x > _____.

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The answer of Consider the function  . We know...

Question 10

A tomato is thrown from the top of a tomato cart its distance from the ground, in feet, is modeled by the equation  where t is measured in seconds and the initial height of the cart is  feet.
(A) At what time is the tomato at its maximum height?
(B) What is the maximum height?
(C) What is the initial velocity of the tomato (at t = 0)?

Accepted Answer

The answer of A tomato is thrown from the top...

Question 11

Let  . Find  .

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The answer of Let  . Find  ....

Question 12

Let  . Find  .

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The answer of Let  . Find  ....

Question 13

A. Find the equation of the line tangent to the graph of  at  .
B. Find the x-intercept of this line.
C. Find the y-intercept of this line.

Accepted Answer

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

Question 14

Find the derivative of 
A) 
B) 
C) 
D)

Accepted Answer

The answer of Find the derivative of 
A) 
B) 
C)...

Question 15

The value of a car is falling at 10% per year so that if  is the purchase price of the car in dollars, its value after t years is given by  . How fast is the car depreciating after 4 years?
A)  dollars per year
B)  dollars per year
C)  dollars per year
D)  dollars per year

Accepted Answer

We can use the formula for exponential decay: 
V(t) = V(0) * e^(-kt) 
where V(t) is the value of the car after t years, V(0) is the initial value (purchase price), k is the decay rate (10% or 0.1), and t is time in years. 
So, we have: 
V(t) = 11eb69ff_fe4f_2a0e_af94_ed09e9f049db_TB8190_11 
V(0) = 11eb69ff_fe4f_02fd_af94_17f75d95dfc8_TB8190_11 
k = 0.1 
t = 4

Substituting these values, we get: 
11eb69ff_fe4f_2a0e_af94_ed09e9f049db_TB8190_11 = 11eb69ff_fe4f_02fd_af94_17f75d95dfc8_TB8190_11 * e^(-0.1*4)

Simplifying, we get: 
e^(-0.4) = 11eb69ff_fe4f_2bdd_af94_5d5c5fb698f2_TB8190_11

Now, we can take the derivative of V(t) with respect to t to find the rate of change: 
dV/dt = -k * V(0) * e^(-kt)

Substituting the values we have, we get: 
dV/dt = -0.1 * 11eb69ff_fe4f_02fd_af94_17f75d95dfc8_TB8190_11 * e^(-0.1*4)

Simplifying, we get: 
dV/dt = 11eb69ff_fe4f_5120_af94_2700f082ef35_TB8190_11

Therefore, the car is depreciating at a rate of 11,696 dollars per year (rounded to the nearest dollar), which corresponds to choice B.

Question 16

Find the derivative of  .
A) 
B) 
C) 
D)

Accepted Answer

The answer of Find the derivative of  .
A) 
B)...

Question 17

Find the derivative of  . 
A)  
B)  
C)  
D)

Accepted Answer

The answer of Find the derivative of  . 
A)...

Question 18

Find the derivative of  .
A) 
B) 
C) 
D)

Accepted Answer

The answer of Find the derivative of  .
A) 
B)...

Question 19

A power function of the form  has  -1/2 and  -1/8. What is n?

Accepted Answer

The answer of A power function of the form ...

Question 20

Find the derivative of  .
A) 
B) 
C) 
D)

Accepted Answer

The answer of Find the derivative of  .
A) 
B)...

Quiz 3: Shortcuts to Differentiation

A landlord rents an apartment building with 250 apartments. The monthly profit, in dollars, can be modeled by , where x is the number of apartments rented. How many apartments should be rented to maximize profit?

Find the derivative of .

Find the equation of the tangent line to the curve which passes through the origin.

Consider the function . We know that is concave down when _ < x < _.

Find the derivative of .

The population of a town is approximated by the function , where t is the number of years since 1980. Find . Round to the nearest whole number.

Find the derivative of

Given and , find .

Consider the function . We know that is increasing when x > _____.

Let . Find .

Let . Find .

A. Find the equation of the line tangent to the graph of at . B. Find the x-intercept of this line. C. Find the y-intercept of this line.

Find the derivative of

The value of a car is falling at 10% per year so that if is the purchase price of the car in dollars, its value after t years is given by . How fast is the car depreciating after 4 years?

Find the derivative of .

Find the derivative of . A) B) C) D)

Find the derivative of .

A power function of the form has -1/2 and -1/8. What is n?

Find the derivative of .