test 4 review chapters 4 and 5 casa testing reservation required…jen/winter mini/notes/filled in...

Math 1431 Test 4 Review 1

Math 1431 Test 4 Review

Chapters 4 and 5

CASA Testing reservation required.

Know your instructor’s name.

Know your lab’s TA name, lab’s time and room number. 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. You will be given scratch paper.

Test: 50 minutes.

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 (8 questions) and FR is worth 36 points (3 questions, with parts).

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

No formulas are provided. No calculators allowed.

REVIEW ALL OPTIMIZATION PROBLEMS FROM SECTION 5.1!!

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


1. If f is invertible and (1) 3, (3) 5, (5) 1,

'(1) 4, '(3) 7, '(5) 2

f f f

f f f

calculate 1 ' 5 .f

Recall: If f (a) = b and '( ) 0f a , then 1 1'( )

'( )f b

f a .

2. Suppose f(x) is an invertible differentiable function and the graph of f passes through the

points (6, -1) and (-1, 2). The slope of the tangent line to the graph of f at x = -1 is 7

2.

Find 1 ' (2)f , the slope of the tangent line to the inverse of f at 2.


3. Find the derivative of each function.

a. 32 4( ) 3 x xf x xe

b. 3ln(2 5) xy x e Then find the equations of the tangent and normal lines at (3, 1).


c. 25 14 xy

d. 2 2

4ln

cos(3 )x

xy

e x x


e. 5logy x

f. y (tan x)(x27)


g. ( ) arcsin(tan( )) arctan( )f x x x

h. ( ) cosh(2 )f x x then find '(0.5)f .


4. Determine where ( ) lnf x x x is increasing/decreasing, any relative extrema, concavity, and

any points of inflection.


5. Use differentials to approximate 4 18 .

Recall: '( )f x h f x f x h

6. Give the differential of 2( ) 3f x x x at x = 1 with increment 110 .


7. You are assigned to design some shipping materials at minimum cost. The package will be a closed rectangular box with a square base, and must have a volume of 50 cubic inches. The material used for the top costs 35 cents per square inch, the material used for the bottom of the box costs 45 cents per square inch, and the material used for the sides costs 20 cents per square inch. Find the dimensions of the box that will minimize the cost. What is the minimum cost?


8. A rectangle is inscribed between the graphs of 2( ) 4f x x and 21( ) 2

2g x x in such a

way that the sides are parallel to the coordinate axes and its upper two vertices are on the f, while the bottom two vertices are on g. What is the largest possible area?


Recall: L’Hospital’s Rule can only be used when you have 0

0 or

.

9. 2

limx

x

e

x

10. 2

0

2sin 1lim

1

x

xx

e x

e x


Indeterminate Forms: 0 , you’ll need to rewrite the expression A B as 1A

B

or 1B

A

.

Reciprocate the one that will be easier to differentiate.

11. 0

lim ln 2x

x x


Indeterminate Powers 00 , 1 and 0 -- Use a logarithmic “trick”. These

indeterminate powers arise if the function is of the form ( )( )

g xf x , where f (x) > 0.

We’ll apply the natural logarithm function.

Indeterminate Forms: Rewrite the function so that you get 0/0 or / .

12. 1/

0lim 1 2

x

xx


13. 7

lim cosx

x x


You decide to completely fence-in your rectangular garden. One side will be adjacent to your neighbor’s property. The neighbor agrees to pay for half of that part of the fence that will border his property. The garden is to be 432 square yards. What dimensions should your garden have so that your cost is at a minimum?

test 4 review chapters 4 and 5 casa testing reservation required…jen/winter mini/notes/filled in...

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