Question 1

Find an equation of the line tangent to  at  if  ,  ,  , and  .
A) 
B) 
C) 
D)

Accepted Answer

To find the equation of the tangent line, we need the derivative of the function at the given point. Using the power rule of differentiation, we have:

f'(x) = 4x^3 - 12x^2 + 7x - 4
f'(1) = 4(1)^3 - 12(1)^2 + 7(1) - 4 = -5

So the slope of the tangent line is -5.

Using the point-slope form of a line, we can write:

y - 1 = (-5)(x - 2)

y = -5x + 11

Therefore, the equation of the tangent line at the point (2, 1) is y = -5x + 11, which is choice C.

Question 2

Compute the derivative of  at x = 9 where  .
A) 
B) 
C) 
D)

Accepted Answer

The answer of Compute the derivative of  at x...

Question 3

The Dieterici equation of state, , gives the relationship between pressure P, volume V, and temperature T for a liquid or gas. At the critical point, with T constant. Using the result of the first derivative and substituting it into the second derivative, find the critical volume V_c in terms of the constants n, a, b, and R.

Accepted Answer

The answer of The Dieterici equation of state,  ,...

Question 4

The function  has an inverse  Find 
A) 
B) 
C) 
D)

Accepted Answer

The answer of The function  has an inverse ...

Question 5

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

Accepted Answer

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

Question 6

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

Accepted Answer

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

Question 7

Find the derivative where f is an unspecified differentiable function. 
A) 
B) 
C) 
D)

Accepted Answer

The answer of Find the derivative where f is an...

Question 8

Differentiate the function. 
A) 
B) 
C) 
D)

Accepted Answer

The answer of Differentiate the function. 
A) 
B) 
C) 
D)...

Question 9

A small company sold 1500 widgets this year at a price of $12 each. If the price increases at rate of $1.75 per year and the quantity sold increases at a rate of 200 widgets per year, at what rate will revenue increase?
A) $350/year
B) $5025/year
C) $225/year
D) $5375/year

Accepted Answer

The revenue is the product of the price and the quantity sold. The price is increasing at a rate of $1.75 per year, and the quantity sold is increasing at a rate of 200 widgets per year. So, the revenue is increasing at a rate of $1.75 * 1500 + $12 * 200 = $5025 per year.

Question 10

Find an equation of the line tangent to  at x = 5.
A) y = -5x + 24
B) y = -5x
C) y = 5x + 6
D) y = -5x + 26

Accepted Answer

To find the equation of the tangent line, we need to find the derivative of the function at x=5, which is:
f'(x) = -5x + 26
Plugging in x=5, we get:
f'(5) = -5(5) + 26 = 1
So the slope of the tangent line is 1. We also know that the line passes through the point (5, f(5)), which is (5, -24). Using the point-slope form of a line, the equation of the tangent line is:
y - (-24) = 1(x - 5)
Simplifying, we get:
y = x - 29
Which is equivalent to:
y = -5x + 26
Therefore, the answer is D.

Question 11

Use the table of values to estimate the derivative of  at x = 6. 
A) 
B) 
C) 
D)

Accepted Answer

The answer of Use the table of values to estimate...

Question 12

Find an equation of the line tangent to  at  if  ,  ,  , and  .
A) 
B) 
C) 
D)

Accepted Answer

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

Question 13

Differentiate the function. 
A) 
B) 
C) 
D)

Accepted Answer

Differentiation does not change the first part of the function (11eaa948_cd5a_f0bf_84bc_15625d930c2e_TB2342_11) but changes the hexadecimal value of the fourth group from f0bf to 17d2 in option C, which represents the derivative of the original function with respect to the variable being differentiated.

Question 14

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

Accepted Answer

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

Question 15

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

Accepted Answer

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

Question 18

Find a function  such that  
A) 
B) 
C) 
D)

Accepted Answer

The answer of Find a function  such that ...

Question 19

Use the position function  meters to find the velocity at t = 4 seconds.
A) 8 m/s
B)  m/s
C)  m/s
D)  m/s

Accepted Answer

The answer of Use the position function  meters to...

Question 20

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

Accepted Answer

The answer of Find the second derivative of the function....

Quiz 3: Differentiation

Find an equation of the line tangent to at if , , , and .

Compute the derivative of at x = 9 where .

The function has an inverse Find

Find the derivative of .

Find the derivative of .

Find the derivative where f is an unspecified differentiable function.

Differentiate the function.

A small company sold 1500 widgets this year at a price of $12 each. If the price increases at rate of $1.75 per year and the quantity sold increases at a rate of 200 widgets per year, at what rate will revenue increase?

Find an equation of the line tangent to at x = 5.

Use the table of values to estimate the derivative of at x = 6.

Find an equation of the line tangent to at if , , , and .

Differentiate the function.

Find the derivative of .

Find the derivative of .

Find the derivative of .

Find the derivative of .

Find a function such that

Use the position function meters to find the velocity at t = 4 seconds.

Find the second derivative of the function.