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MATH 1013 SECTION A: Professor Heffernan APPLIED CALCULUS I, FALL 2010 SECTION B: Professor Szeto NAME: STUDENT #:

Final Exam 13 December 2010, 19:00-22:00 No other aids except a non-graphing calculator is allowed. ANSWER AS MANY QUESTIONS AS YOU CAN. All questions carry equal marks. In all questions it is essential to explain your reasoning and to provide details of the intermediate steps taken in reaching your answers. Answers are to be written in this question book. Do not remove any pages. IF you need extra space for your answers use the backs of pages, but CLEARLY indicate which answer is which! Make sure to write your name and student number on the paper. Students will not be allowed to leave during the final 15 minutes of the examination period in order to avoid disruption to those continuing to work on the paper. MARKING TEMPLATE

1 2 3 4 5 6 7 8 9 10 TOTAL

Question 1. Find the natural domains and ranges of the functions: a) g(x) = tan(π-x) b) f(x) = 1/(4+ex) c) h(x) = 1/(4 - ex)

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Question 2. Find each of the following limits OR indicate that it does not exist. If the limit does not exist clearly explain why not.

a)

!

lim" # 0

1$ cos"

(sin" )2 b)

!

limt" 64

t # 8

64 # t

c)

!

limx" 2

2# x

| x # 2 | d)

!

limu"#

(u2

+ 4u

u2$1

)

Question 3. The function f(x) = x - 2x

refers to all part questions below.

a) Where is the function continuous? Where is the function not continuous? b) Consider the following statement:

“f(-2) = -1, f(2) = 1. Therefore the Intermediate Value Theorem assures us that there must be at least one value of c between -2 and 2, such that f(c)=0.” Do you agree with the statement? If you do not, explain why not. Required: begin by explaining the IVT.

c) If you believe that the Mean Value Theorem applies to f(x) given above on the interval [-2,2], find one value of x between -2 and 2 that satisfies the theorem. If you believe that the theorem does not apply, explain why not. Required: begin by explaining the MVT.

Question 4. a) Find the indicated derivatives using any method you like. Simplify as far as possible.

(i)

!

d

dt[tan(cos

23t)] (ii)

!

d

dx(ln(3e

4x))

b) Use the limit definition of a derivative (any other method will receive no merit) to obtain

the following derivative

!

d

dx(1

x + 3)

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Question 5. a) Find dy/dx by implicit differentiation (or any other valid method) if

!

(x2

+ y2)2

= 92

b) At what values of x, if any, is dy/dx undefined? Question 6. Make an analysis (natural domain, range, symmetry if any, asymptotes if any, increasing/decreasing regions, relative and absolute max and min if any, concave up/down

regions, points of inflection if any, etc.) of the function

!

y(x) = e

1

x . Sketch the curve, labelling axes, intercepts if any, asymptotes if any, points of max and min if any, and points of inflection if any. Clearly indicate intervals of x where the function is increasing or decreasing, and the curve is concave up or concave down. Question 7. An isosceles triangle (i.e. a triangle with two equal sides) BCG circumscribes a circle of radius 1m, as shown in the diagram. Angle θ has been introduced for convenience of reference, as well as several points B,C,D,E,F,G. Let A denote the area of triangle BCG.

a) Show that A is related θ via

!

A =(1+ sin" )

2

cos" sin".

[Hint: work out lengths CD and FG in terms of θ by considering appropriate triangles CDE and CFG.]

b) Show that A is minimized if

!

" = # / 6 . [You are required to demonstrate that a minimum rather than a minimum is reached by applying an appropriate test. First derivative test is recommended since it is easier.]

θ θ

1 1

C

D E

F G B

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Question 8. Consider the function f(x) = x2 + 1 over the interval [1,4]. a) Write out and evaluate the Riemann Sum R3 obtained by partitioning [1,4] into 3 equal

sub-intervals and choosing sample points

!

xi

*= x

i (i.e. the right hand end of the i-th

interval). [Hint, in terms of terminology used in your text, figure out a, b, n, Δx, x1, x2, x3. Proceed from there.]

b) Generalize to N equal sub-intervals, and write an expression for RN over the same interval. You are expected to use “sigma” notation to express the required sum. To evaluate the sum, you may assume

!

i

i=1

n

" =n(n +1)

2, i

2

i=1

n

" =n(n +1)(2n +1)

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c) Evaluate

!

limN"#

RN

Question 9.

a) Use L'Hospital's rule to determine

!

limx" 0

(sin x) # x

x3 .

b) Evaluate and simplify as far as possible:

i)

!

x4 "1

x5# dx ii)

!

(et " e"t )2# dt iii)

!

(3

cos2"

# 2sin" ) d"0

$4%

Question 10. Using any method you like, evaluate the following and simplify as far as possible:

a)

!

d

dx 1

x3

" arcsinu du b) ⌡⌠

0

π/2 (cos x) (sin(sin x)) dx

End of Exam