3 problems on mean value theorem 6 problems on derivatives and graphs *see attachments *solve the problems on calculus I level

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MATH 211

Learning Activity – Mean Value Theorem

Fall 2017

Name:

1. Determine whether or not Rolle’s Theorem applies to the following functions on the given intervals.

If it does, find the point(s) that are guaranteed to exist by Rolle’s Theorem.

(a) f (x) = 1 − x2/3 on [−1, 1].

h πi

(b) f (x) = sin(2x) on 0, .

2

MATH 211

Learning Activity – Mean Value Theorem

Fall 2017

2. Verify that the function f (x) = x3 + x − 1 satisfies the Mean Value Theorem on the interval [0, 2].

Then, find all numbers c guaranteed to exist by the Mean Value Theorem.

MATH 211

Learning Activity – Derivatives and Graphs

Fall 2017

Name:

1. Consider the function f (x) = x2 + 3 on the interval [−3, 2].

(a) Find the critical points for the function.

(b) Use the First Derivative Test to locate the local maximum and minimum values. Report both

the x and the y coordinates.

(c) Identify the absolute maximum and minimum values for the function on the given interval.

Again, report both the x and the y coordinates.

MATH 211

Learning Activity – Derivatives and Graphs

2. Consider the function f (x) = x2/3 (x − 4).

(a) Find the critical points.

(b) Find where the function is increasing and decreasing.

Fall 2017

MATH 211

Learning Activity – Derivatives and Graphs

3. Suppose you are given a function f (x) and you discover that the first derivative is

f 0 (x) = x3 (x − 5)5 (x + 4)2 .

Determine all the points where f (x) has local extreme values.

Fall 2017

MATH 211

Learning Activity – Derivatives and Graphs (Continued)

Name:

1. Consider the function f (x) =

√

3

x − 4.

(a) Determine the intervals on which the function is concave up and down.

(b) Identify any inflection points. Report both the x and the y coordinates.

Fall 2017

MATH 211

Learning Activity – Derivatives and Graphs (Continued)

Fall 2017

2. Locate the critical points of the following function and use the Second Derivative Test to determine

whether they correspond to local minima, local maxima, or neither. Report both the x and y

coordinates.

f (t) = 2t3 + 3t2 − 36t

MATH 211

Learning Activity – Derivatives and Graphs (Continued)

3. Suppose you are given a function f (x) and you discover that the second derivative is

f 00 (x) = 5(x2 − 1)2 (x + 3)(x − 2)6 .

Determine all the x-values at which f (x) has inflection points.

Fall 2017

…

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