Description

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1.

Determine the value of each of the following limits (if they exist).

Use the method of your choice, be sure to show all work. If you are using a theorem from your textbook,

then you must state it by name.

(8pt)

πππ

a.

π₯ββ

πππ

b.

π₯ 2 β3π₯+2

3π₯β8

π₯ 2 β3π₯+2

π₯β2 2π₯ 2 β8

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2.

Neatly and carefully construct a coordinate grid (such as provided below). On your coordinate grid, sketch

a graph of a function π¦ = π(π₯) that satisfies all the following conditions:

(10pt)

a. lim π(π₯) = 2

π₯β5

b.

c.

lim π(π₯) = β4

π₯ββ

lim π(π₯) = 6

π₯β β5β

d.

e.

lim π(π₯) = β

π₯β β5+

lim π(π₯) = β

π₯βββ

3.

Use the following graph of a function, π(π₯) to answer each of the following multiple-choice questions.

(12pt)

I.

The derivative, πβ²(π₯), is equal to 0 when

a. x = -3, 1

II.

d. x = 0

b. (0.7, 4)

c. (-2.8, 4)

d. (-1, 2)

The derivative, πβ²(π₯), is increasing on the interval(s)

a. (-3, 3)

IV.

c. x = -1, 2

The derivative, πβ²(π₯), is positive on the interval(s)

a. (-5, -1)

III.

b. x = -5, 0.7, 4

b. (-5, 1)

c. (-1, 2)

d. (2, 5)

c. x = -1, 4

d. x = -2

The 2nd derivative, πβ²β²(π₯), is equal to 0 when

a. x = 1

b. x = -5, 0.7, 4

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3

4.

Let π¦ = βπ₯ 5 . se the power rule to find π¦β². Show detailed steps.

(5pt)

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5.

Let π¦ = (2 ln π₯)3 .

(5pt)

a.

The given π¦ is a composition of functions π(π(π₯)) where

π(π₯) = ________________ and π(π₯) = _________________ .

b.

Show detailed steps how you would find π¦β²

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6.

The value of functions π(π₯) and π(π₯), as well the value of their derivatives πβ²(π₯) and πβ²(π₯)

(16pt)

are provided for some specific x values in the table below:

π₯

π(π₯)

πβ²(π₯)

π(π₯)

πβ²(π₯)

2

1

2

0

-1

4

-2

4

2

-2

6

-1

1

3

-2

8

4

-2

4

-1

Use the above table to determine each of the following:

a.

π(π₯) = π(π₯) β π(π₯). Find πβ²(4).

b.

π(π₯) = π(π₯)π(π₯). Find πβ²(6).

Note: the bonus problem at the end of the exam is related to this question.

c.

π(π₯) = π(π(π₯)). Find πβ²(8).

d.

π(π₯) = [π(π₯)]β1 . Find πβ²(2).

7.

Application of Implicit Differentiation

(16pt)

A hot air balloon is rising vertically at a rate

of 100 meters per minute.

A camera located 50 meters away from the

balloon launch site is filming the balloon.

Note that as the balloon rises, the angle

between the camera and the ground must be

increasing.

Your task:

Determine the rate at which the angle, π, is increasing when the balloon is 150 meters above the ground.

a.

How is the angle, π, is related to the height of the balloon? (Based on the picture; think trig).

b.

Use implicit differentiation to differentiate your answer to part a. with respect to time.

ππ

Solve for .

ππ‘

Note: your result should have a

πβ

ππ‘

term and furthermore, you have been given that

πβ

ππ‘

= 100.

c.

Determine the angle of the camera when the balloon is 150 meters off the ground.

d.

Use your results to b. and c. to determine the rate at which the angle, π, is changing when that

balloon is 150 meters off the ground.

8.

Given π(π‘)

= 2βπ‘ ,

(15pt)

a.

Calculate πβ²(π‘) then evaluate πβ²(1)

b.

Calculate πβ²β²(π‘) then evaluate πβ²β²(1)

c.

At π‘ = 1 is π increasing, decreasing, or neither?

What information from your answers to a. or b. lead you to this conclusion?

d.

At π‘ = 1 is π concave up, concave down, or neither?

What information from your answers to a. or b. lead you to this conclusion?

e.

At π‘ = 1 is πβ² increasing, decreasing, or neither?

What information from your answers to a. or b. lead you to this conclusion?

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9.

Consider the curve given by the equation π₯ 4 + π¦ 3 = 3π₯ 2 π¦.

(10pt)

a.

ππ¦

Use implicit differentiation and find an equation for ππ₯ .

(Your final answer may contain both x and y)

b.

Give the equation of the tangent line to this curve at the point (2,2).

10.

(3pt)

One of the following is the graph of a function, π(π₯), another is the graph of itβs derivative, πβ²(π₯), and

another is the graph of itβs 2nd derivative πβ²β²(π₯).

Identify which graph is π(π₯), which graph is πβ²(π₯), and which graph is πβ²β²(π₯).

8

6

4

2

0

-2 0

-4

-6

4

6

2

4

2

0

2

4

6

-2

8

0

2

4

6

8

0

-4

,

Graph 1

-2

-6

,

0

2

4

-4

Graph 2

Bonus: (worth +3)

Consider problem 6b. Provide the equation of the tangent line to π(π₯) when π₯ = 6.

Graph 3

6

8

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