Question 1

Sketch a graph with the following conditions:  and  .

Accepted Answer

The answer of Sketch a graph with the following conditions:...

Question 2

Estimate the value of  for the function  .

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The answer of Estimate the value of  for the...

Question 3

A table of values is given for  .
A. Is  positive or negative?
B. Is  positive or negative?
C. Approximate  by averaging the approximations from either side.

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The answer of A table of values is given for...

Question 4

In 2007, Apple's iTunes music store sold 2 billion songs. The number of iTunes songs purchased (in millions) is shown on the following chart, S(t), where time is measured in days since Apple iTunes sold 1 million songs (March 15, 2003).  
A) Estimate
  with the appropriate units. 
B) Use
  to estimate
  to 2 decimal places.

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The answer of In 2007, Apple's iTunes music store sold...

Question 5

Given the following data about the function, f, use an approximation of the tangent line at x=4.5 to estimate  .

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The answer of Given the following data about the function,...

Question 6

Assume that f and g are differentiable functions defined on all of the real line. f can satisfy:  everywhere,  everywhere, and  everywhere.

Accepted Answer

This is true by the chain rule of differentiation.

Question 7

The function  is graphed below. Which is larger,  or  ?

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The answer of The function  is graphed below. Which...

Question 8

There is a function used by statisticians, called the error function, which is written y=erf(x). Suppose you have a statistical calculator, which has a button for this function. Playing with your calculator, you discover the following:  Using this information alone, give an estimate for erf '(0), accurate to 2 decimal places.

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The answer of There is a function used by statisticians,...

Question 9

The function  is graphed below. Which is larger,  or  ?

Accepted Answer

The larger value is the maximum point on the graph. In this case, it is 11eb69ff_fe8e_07c8_af94_a5538e7630c0_TB8190_11. Choice B is not larger than choice A.

Question 10

Assume that f and g are differentiable functions defined on all of the real line. It is possible that  everywhere,  everywhere, and  everywhere.

Accepted Answer

If f(x) and g(x) are both differentiable functions and their derivatives are always equal, then f(x) and g(x) differ by at most a constant. Therefore, it is not possible for f(x) and g(x) to be different functions with equal derivatives everywhere.

Question 11

The function  is graphed below.
A. Is  positive, negative, or zero?
B. Is  positive, negative, or zero?

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The answer of The function  is graphed below.
A. Is...

Question 12

Assume that f and g are differentiable functions defined on all of the real line. If  for all x and if  for some  , then  for all x.

Accepted Answer

This is a direct application of the Mean Value Theorem for differentiation. Since f and g are differentiable functions, we can apply the Mean Value Theorem to both of them.

For some value c, we have:

f(b) - f(a) = f'(c)(b-a)

g(b) - g(a) = g'(c)(b-a)

We are given that f(b) - f(a) = g(b) - g(a). Therefore, we have:

f'(c)(b-a) = g'(c)(b-a)

Since (b-a) ≠ 0 (by assumption), we can divide both sides by (b-a) to obtain:

f'(c) = g'(c)

Since this holds for all c in the interval (a,b), it must be true that f'(x) = g'(x) for all x in the real line (by continuity of f' and g'). Now, we can integrate both sides with respect to x to get:

f(x) = g(x) + C

where C is an arbitrary constant. But since we are given that there exists a value x_0 such that f(x_0) = g(x_0), we can solve for C:

C = f(x_0) - g(x_0)

Substituting back into the equation f(x) = g(x) + C, we obtain:

f(x) = g(x) + (f(x_0) - g(x_0))

This holds for all x, so we have shown that f(x) - g(x) is a constant function.

Question 13

Which point has a slope of -0.5 ?

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The answer of Which point has a slope of -0.5...

Question 14

The following table gives the number of passenger cars, in millions, in the United States, C, as a function of years, t. We have  . Estimate  . Use the nearest right-hand value to make your estimate.

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The answer of The following table gives the number of...

Question 15

The first figure shows the graph of the derivative of a function. Could the second figure be the original function?

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The answer of The first figure shows the graph of...

Question 16

There is a population of  thousand bacteria in a culture at time t hours after the beginning of an experiment. You know that  ,  , and  . Using these values, make a prediction for

Accepted Answer

We know that the population is increasing exponentially, so we need to use the formula:

N(t) = N0 * e^(rt)

where N0 is the initial population, r is the growth rate, and t is the time.

Using the given values, we have:

N0 = 11
N(4) = 27
N(8) = 81
N(12) = ?

To solve for r, we can use the fact that:

N(t2)/N(t1) = e^(r(t2-t1))

Using t1 = 4 and t2 = 8, we have:

N(8)/N(4) = e^(r(8-4))
81/27 = e^(4r)
3 = e^(4r)
ln(3) = 4r
r = ln(3)/4

Now we can use this value of r to find N(12):

N(12) = N0 * e^(r*12)
N(12) = 11 * e^(ln(3)/4 * 12)
N(12) = 11 * e^3
N(12) = 11 * 20.0855
N(12) = 220.9405

Therefore, the predicted population at 12 hours is 220.9405 thousand, which rounds to 0.304 million. So the answer is B.

Question 17

There is a population of  thousand bacteria in a culture at time t hours after the beginning of an experiment. You know that  ,  , and  . Using these values, make a prediction for  .

Accepted Answer

The answer of There is a population of  thousand...

Question 18

The following table gives the number of passenger cars, in millions, in the United States, C, as a function of years, t. We have  . Is f ''(t) positive or negative?

Accepted Answer

The answer of The following table gives the number of...

Question 19

Given the following data about the function, f, use estimates of  and  to estimate  . Use the nearest right-hand value to make your estimate.

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The answer of Given the following data about the function,...

Question 20

Assume that f and g are differentiable functions defined on all of the real line. f and g can satisfy:  for all x and  for all x.

Accepted Answer

This statement is true by the product rule of differentiation, which says that (fg)' = f'g + fg'. Since the statement given in the question says that (fg)' = f'g + fg, it follows that f'g = gf', which is satisfied by functions of the form f(x) = Ae^kx and g(x) = Be^kx, where A and B are constants and k is a constant that is the same for both functions.

Quiz 2: Rate of Change: the Derivative

Sketch a graph with the following conditions: and .

Estimate the value of for the function .

A table of values is given for . A. Is positive or negative? B. Is positive or negative? C. Approximate by averaging the approximations from either side.

Given the following data about the function, f, use an approximation of the tangent line at x=4.5 to estimate .

Assume that f and g are differentiable functions defined on all of the real line. f can satisfy: everywhere, everywhere, and everywhere.

The function is graphed below. Which is larger, or ?

The function is graphed below. Which is larger, or ?

Assume that f and g are differentiable functions defined on all of the real line. It is possible that everywhere, everywhere, and everywhere.

The function is graphed below. A. Is positive, negative, or zero? B. Is positive, negative, or zero?

Assume that f and g are differentiable functions defined on all of the real line. If for all x and if for some , then for all x.

Which point has a slope of -0.5 ?

The following table gives the number of passenger cars, in millions, in the United States, C, as a function of years, t. We have . Estimate . Use the nearest right-hand value to make your estimate.

The first figure shows the graph of the derivative of a function. Could the second figure be the original function?

There is a population of thousand bacteria in a culture at time t hours after the beginning of an experiment. You know that , , and . Using these values, make a prediction for

There is a population of thousand bacteria in a culture at time t hours after the beginning of an experiment. You know that , , and . Using these values, make a prediction for .

The following table gives the number of passenger cars, in millions, in the United States, C, as a function of years, t. We have . Is f ''(t) positive or negative?

Given the following data about the function, f, use estimates of and to estimate . Use the nearest right-hand value to make your estimate.

Assume that f and g are differentiable functions defined on all of the real line. f and g can satisfy: for all x and for all x.