Name:

Elementary Functions and Calculus II Math 132 (Sec 22), Autumn 2004 Mid-term 1

30th January 2004

Instructions: The total time allowed for this examination is 50 minutes. The use of notes, textbooks or calculators is prohibited. Write you answer to each question in the space provided below it. Should you require more space write on the reverse of the paper, labeling your answers clearly. Write your name at the top of this sheet. The maximum number of points available for each question or part question is shown in parentheses next to the question. There are 4 questions. You should attempt as many of the questions as you can. Partial credit may be given for incomplete answers.

1. The figure shows a unit circle and a line joining the origin to the point (x, y). The length of the arc from (1, 0) to (x, y) is denoted by θ.

(x, y) (1, 0) θ

(a) (6 pts) On the figure, label the lengths which define cos θ and sin θ. Also give the definition of tan θ below.

Math 132 (Sec 22)  Mid-term 1  30th January 2004

(b) (7 pts) Use the identity cos(θ + ϕ) = cos θ cos ϕ − sin θ sin ϕ to derive an identity for cos(2t).

(c) (7 pts) Use another identity you know to write cos(2t) in terms of powers of cos t only. State clearly the identity you are using.

Math 132 (Sec 22)  Mid-term 1  30th January 2004

2. (a) i. (5 pts) In lectures we proved that lim_{x→0}(sin x)/x and lim_{x→0}(1−cos x)/x exist. State (without proof) the values we computed them to be.

ii. (5 pts) Since all the functions involved in the limits above are continuous, why can we not just plug in the value x = 0 to calculate the limit? Be careful to state the exact reason.

(b) Calculate the following limits using the results in part (a) and other standard results you

know. Label clearly points at which you use these facts.

i. (5 pts)

lim_{x→0} \frac{tan x}{x}

Math 132 (Sec 22)  Mid-term 1  30th January 2004

ii. (5 pts)

lim_{x→0} \frac{cos θ − cos(θ + x)}{x}

3. Define f : [0, 4] → R by f(x) = 4x − 3 − x^{2} for all x ∈ [0, 4].

(a) (10 pts) Find all the critical points of f. Page 4 of 6 (b) (5 pts) What conditions do f and its domain satisfy that justify the existence of a (global) max-

imum value and (global) minimum value of f on [0, 4].

(c) (5 pts) Find the (global) minimum value of f on [0, 4]. How do you know this is the minimum value?

4. (a) (10 pts) State the mean value theorem.

(b) (10 pts) Use the mean value theorem to show that if f : [a, b] → R is differentiable on (a, b), continuous on (a, b) and f′(x) = 0 for all x ∈ (a, b) then there exists a constant D such that f(x) = D for all x ∈ [a, b].