5.1 AB MVT

Multiple Choice

Given $$x+3=x^2y$$ , find $$\frac{dy}{dx}$$ .

• $$\frac{1}{x^2-2xy}$$

• $$\frac{1}{x^2+2xy}$$

• $$\frac{1-2xy}{x^2}$$

• $$\frac{2xy+1}{x^2}$$

Multiple Choice

Let g be a continuous function on the closed interval [-1,4], where g(-1)=-4 and g(4)=1.

Which of the following is guaranteed by the Intermediate Value Theorem (IVT)?

• g(c)=-3 for at least one c between -4 and 1

• g(c)=3 for at least one c between -1 and 4

• g(c)=3 for at least one c between -4 and 1

• g(c)=-3 for at least one c between -1 and 4

Multiple Choice

Let $$f\left(x\right)=x^2-4x+5$$ . Find the average rate of change of f(x) over the interval [1,5].

• 4

• 2

• 8

• 1/4

Multiple Choice

Let $$f\left(x\right)=x^2-4x+5$$ . Find the instantaneous rate of change of f(x).

• $$f'\left(x\right)=2x$$

• $$f'\left(x\right)=2x-9$$

• $$f'\left(x\right)=2x-4$$

• $$f'\left(x\right)=2x+1$$

Multiple Choice

$$f\left(x\right)=x^2-4x+5$$

Is there a value of x where the instantaneous rate of change of f(x) equals the average rate of change of f(x) over the interval [1,5]?

• none

• x=3

• x = -1

• x=1

Draw

For each function below, draw the segment between the indicated points on the graph. Is there a point on the graph between the indicated points where the slope of the tangent line is equal to the slope between the given points? If so, sketch the tangent line.

Draw

For each function below, draw the segment between the indicated points on the graph. Is there a point on the graph between the indicated points where the slope of the tangent line is equal to the slope between the given points? If so, sketch the tangent line.

Match

Match the following characteristics of the IVT versus MVT.

• IVT Conditions:

• MVT Conditions:

• Conclusion to draw in IVT problems:

• Conclusion to draw in MVT problems:

Continuity

Continuity & Differentiability

Values of the function

Rate of change of the function

• Continuity

• Continuity & Differentiability

• Values of the function

• Rate of change of the function

Vicki drove 150 miles in 3 hours. The average rate of change of her distance traveled with respect to time was 50 miles per hour.

The Mean Value Theorem tells us there must have been at least one moment in time, between noon and 3:00 pm, when she was traveling exactly 50 mph.

​Determine whether you would use the IVT or MVT to answer each question.

​Since v(t) is differentiable on the interval (0,9), it must be continuous. The Intermediate Value Theorem guarantees that there is a time c, for 0<c<9, such that v(c)=-1 because v(c)=-1 is between v(9)=-4 and v(0)=5.