Question 1

Suppose that  is a function such that  . What is  ?

Accepted Answer

The answer of Suppose that  is a function such...

Question 2

Given a function  , what does the quantity  represent?

Accepted Answer

The slope of the secant line to the graph of  containing the points on the graph of  where  and  .

Question 3

If  then

Accepted Answer

E

Question 4

An equation of the tangent line to the graph of  at both  and  is  . Which of the following statements must be true?

Accepted Answer

Since the line equation is given as the equation of the tangent line at two points on the graph of the function, it must be the derivative of the function at those points. Therefore, the function must be differentiable at those points, and option D cannot be true. The line equation matches the form of the derivative of the function 11eaa655_3b44_3cf9_a696_abe4e84538f4_TB5596_11 at those points, so option C is true. Option A is not necessarily true as the function could be another linear function that also passes through those two points. Option B is not necessarily true as we do not have enough information about the concavity of the function. Therefore, the correct answer is C.

Question 5

True or False: The graph of the function  has a vertical tangent line at  .

Accepted Answer

The answer of True or False: The graph of the...

Question 6

What is the derivative of the function  at the point where  ?

Accepted Answer

The answer of What is the derivative of the function...

Question 7

Let  be a function satisfying  . Which of the following statements must be true?

Accepted Answer

Since the function satisfies the condition 11eaa655_3b45_2776_a696_216c23381c27_TB5596_11, it must be continuous at every point. If the tangent line to the graph of the function at a point has a positive slope, it means that the function is increasing at that point. Since the function is continuous, if it is increasing at one point, it must be increasing in a small interval around that point. Therefore, the statement C must be true. The other statements do not follow from the given information.

Question 8

Let  . Compute  and find an equation of the tangent line to the graph of  at

Accepted Answer

The answer of Let  . Compute  and find...

Question 9

If  is a variable and  is a constant, which of the following expressions equals  ?

Accepted Answer

The answer of If  is a variable and ...

Question 10

Which of the following functions is continuous at  but not differentiable there?

Accepted Answer

The function 11eaa655_3b45_9bb6_a696_4176db03b1d8_TB5596_11 is the only option that has the same constant coefficient (11) as in the given point, and a non-integer exponent, which means it is not differentiable at that point (since the derivative of a function with a non-integer exponent is not defined at points with negative values). Furthermore, it is continuous at that point because the limit of the function as x approaches 11eaa655_3b45_74a3_a696_0f7ea9c37f07_TB5596_11 is equal to its value at that point, which is also 11eaa655_3b45_9bb6_a696_4176db03b1d8_TB5596_11. Therefore, the correct answer is C.

Question 11

If  what is  ?

Accepted Answer

The answer of If  what is  ?...

Question 12

What does the limit  represent?

Accepted Answer

The answer of What does the limit  represent?...

Question 13

What is the derivative of  ?

Accepted Answer

The answer of What is the derivative of  ?
...

Question 14

We can use the difference quotient  as an approximation to  when  and  are known. 
A) Find an approximation to  based on the values of  and  .
Given  such that:  
B) Use the approximation found in (A) to approximate  
C) Use the approximation found in (A) to approximate

Accepted Answer

The answer of We can use the difference quotient ...

Question 15

Suppose  on  . If  ,  ,  ,  , and  , what is  ?

Accepted Answer

The answer of Suppose  on  . If ...

Question 16

The derivative of the function  at the point where  equals

Accepted Answer

The answer of The derivative of the function  at...

Question 17

True or False: The derivative of  is  .

Accepted Answer

The answer of True or False: The derivative of ...

Question 18

Find the  coordinates of all the points on the curve  , besides  , at which the slope of the tangent line equals 1.5.

Accepted Answer

The answer of Find the  coordinates of all the...

Question 19

Find  and  if an equation of the tangent line to the graph of the function  at  is  .

Accepted Answer

The answer of Find  and  if an equation...

Question 20

At which points is the function shown in the figure not differentiable?

Accepted Answer

The answer of At which points is the function shown...

Quiz 3: Differentiation

Suppose that is a function such that . What is ?

Given a function , what does the quantity represent?

If then

An equation of the tangent line to the graph of at both and is . Which of the following statements must be true?

True or False: The graph of the function has a vertical tangent line at .

What is the derivative of the function at the point where ?

Let be a function satisfying . Which of the following statements must be true?

Let . Compute and find an equation of the tangent line to the graph of at

If is a variable and is a constant, which of the following expressions equals ?

Which of the following functions is continuous at but not differentiable there?

If what is ?

What does the limit represent?

What is the derivative of ?

We can use the difference quotient as an approximation to when and are known. A) Find an approximation to based on the values of and . Given such that: B) Use the approximation found in (A) to approximate C) Use the approximation found in (A) to approximate

Suppose on . If , , , , and , what is ?

The derivative of the function at the point where equals

True or False: The derivative of is .

Find the coordinates of all the points on the curve , besides , at which the slope of the tangent line equals 1.5.

Find and if an equation of the tangent line to the graph of the function at is .

At which points is the function shown in the figure not differentiable?