test 3 review material covered: chapter 3jen/winter mini/notes/filled in notes/ch 3... · math 1431...

Math 1431 Test 3 Review 1

Math 1431 Test 3 Review

Material Covered: Chapter 3

Know your instructor’s name.

Know your lab’s TA name, lab’s time and room number.

Test: 50 minutes; CASA Testing reservation required.

Test is part multiple choice and part free response (with parts). You will have the multiple choice grade right away, but the free response will take some time to grade.

The multiple choice is worth 64 points and FR is worth 36 points.

The related rates problem on the FR, there will be two and you will choose only ONE to do.

No formulas are provided. No calculators allowed.

You will be given scratch paper.

For the free response you’ll be given stapled pages asking for your name, lab info, etc to do all your free response work on. Show all your work here and not on the scratch paper.

Do the practice test (20 attempts allowed) for more practice and extra credit.


1. On a morning when the sun will pass directly overhead, the shadow of a 60-ft tower on level ground is 45 feet long. At the moment in question, the angle the sun makes with the ground is increasing at the rate of π/600 radians/minute. Find the rate of change of the shadow's length. Picture:

Given: Question:

Relation:


2. A pile of sand in the shape of a cone whose radius is twice its height is growing at a rate of 5 cubic meters per second. How fast is its height increasing when the radius is 20 meters? Picture:

Given: Question:

Relation:


3. A particle moves on the number line with position function: 3 26 1s t t t . Is the

object speeding up or slowing down at t = 3?

4. A stone is thrown straight up with an initial velocity of 128 feet per second, so that its height (in feet) after t seconds is given by 2( ) 128 16h t t t . When will the ball hit the ground? At what time is the velocity zero?


5. Verify that the Rolle’s Theorem applies to the function 2f x cos x over 0, . Then

find all points in this interval that satisfy Rolle’s Theorem.

If f continuous on 0, ?

If f differentiable on 0, ?

f(0) =

f( ) =


6. The conclusion of the Mean Value Theorem states that there exists a point c in the interval (a, b) such that the tangent line is parallel to the line passing through (a, f(a)) and (b, f(b)).

Let xxxf 3)( 3 be defined on [-1, 1]. Find c on (-1, 1) that satisfies the conclusion of the Mean Value Theorem.

If f continuous on [-1, 1]? If f differentiable on (-1, 1)?


7. The graph of '( )f x is given. Assume the original function’s domain is all real numbers.

a. Give the x-values of any critical numbers of f.

b. Over which intervals is f increasing? Decreasing?

c. Give any x-values of extreme points of f.

d. Over which intervals is f concave up? Concave Down? Recall: Whenever f ‘ is increasing, f is concave up. Whenever f ‘ is decreasing, f is concave down.

e. Any points of inflection for f ?


8. The graph of ''( )f x is given. Assume the original function’s domain is all real numbers.

a. Over which intervals is f concave up? Concave Down?

b. Find x-values of any points of inflection for f.


9. Find any critical numbers and classify them as a local maximum, local minimum or neither for:

a. 3

2 5f 3x x x ; does it have any vertical cusps or vertical tangents?


b. Find any critical points and classify them as local min/max on 2

0,3

.

2 2f x sin x cos x ,


10. Find the absolute extrema of the function 2

4 4( )

xf x

x

defined on the interval

[1, 8].


11. Let 2

10( )

( 4)

xf x

x

. State the critical numbers, the intervals where the function is

increasing/decreasing, concave up/down, and graph this function. Carefully label the x and y-intercepts, local extrema, and point(s) of inflection. Domain is all real numbers except 4.

Asymptotes: lim ( ) 0x

f x

; ( )f x as x 4 and as x 4

Intercepts: (0) 0f

3

10 40'( )

( 4)

xf x

x

;

Find when '( ) 0f x .

Critical Points:

Find when '( )f x is undefined.

( )f x : Increasing: Decreasing:

'( )f x : Rel Ext:


4

20 160''( )

( 4)

xf x

x

Find when ''( ) 0f x .

Find when ''( )f x is undefined.

( )f x : Concave Up: Concave Down:

''( )f x : P.O.I.:


12. Sketch the graph of a function that has: Vertical Cusp at x = 1, Vertical Tangent at x = 2, Vertical Asymptotes at x = -2 and x = 3, and lim ( ) 3

xf x

.

13. Given the following “sign chart” of both the first and second derivatives of a function f with domain all real numbers.

x 5x x = -5 5 0x x = 0 0 3x x = 3 3x

'( )f x Negative 0 Positive 4 Positive 0 Positive

''( )f x Positive 3 Positive 0 Negative 0 Positive

a. Find any critical numbers.

b. Give any relative extrema.

c. Give any points of inflection.


Make sure to work through all other related rates from Section 3.1! Two cars are moving towards the same point. The first car started from a point that is 100 miles away and it travels south at a rate of 40 mi/h. The second one started from a point 105 miles away and it travels east at a rate of 60 mi/h. At what rate is the distance between the cars changing one hour later? The altitude of a triangle is increasing at a rate of 5 in/min while the base is decreasing at a rate of 1 in/min. At what rate is the area of the triangle changing when the altitude is 10 in and the base is 6 in?

Water is poured into a conical tank at the rate of 64 cubic feet per minute. If the tank is 10 feet tall and the top of the tank has a radius of 6 feet, how fast does the radius of the water surface change when the water is 2 feet deep?

test 3 review material covered: chapter 3jen/winter mini/notes/filled in notes/ch 3... · math 1431...

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