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MATH 150
Final Examination, December 17, 2005 R. Pyke
Last Name:
First Name:
SFU Student ID :
- DO NOT LIFT UP THE COVER PAGE UNTIL INSTRUCTED.
- Once the test begins, please check that all pages are intact.
- If the answer space provided is not sufficient, write your answer on the back of the previous page. Clearly mark the question number.
- Ordinary Scientific Calculators ONLY are allowed. NO GRAPHING CALCULATORS ALLOWED.
- Duration of exam: 180 minutes. Marks for each question are indicated by [ ].
Some formulae:
sin^2 θ + cos^2 θ = 1, tan^2 θ + 1 = sec^2 θ, 1 + cot^2 θ = csc^2 θ
sphere : V =
πr^3 , S = 4πr^2
circle : A = πr^2 , C = 2πr
triangle : A =
bh
Question Mark Question Mark
6 /7 Total /
(1) [8] Find the indicated limits. If the limit does not exist explain why.
(1a) lim x→− 2
x + 2 |x^2 − x − 6 |
(1b) lim z→ 0
4 tan 3z sin 2z
(1c) lim
x→ 0 +
(e
2 x
+ x)
1 x
(3) [5] A ball is dropped from a height of 200 metres, 50 metres from a lamp that is also 200 metres tall. The ball’s height above the ground at time t is s(t) = 200 − 10 t^2 metres. Determine how quickly the ball’s shadow (due to the lamp) is moving along the ground when the ball is half way to the ground.
(4) [4] Consider the following function;
f (x) =
{ cx + b x < 0 3 + (x − 2)^2 x ≥ 0
Determine the values of c and b, if there are any, so that f (x) is a differentiable function for all x.
(5) [4] Suppose f (x) is a differentiable function on the interval [1, 6]. Suppose too that f (5) = −1 and |f ′(x)| ≤ 32 for all x in [1, 6]. Use the Mean Value Theorem to determine all possible values of f (1) and f (6).
(7) (a) [2] Use implicit differentiation to show that
d dx
( tan−^1 x
)
1 + x^2
(Hint: Consider the equation tan y = x.)
(b) [2] Derive the equation of the linearization L(x) of a function f (x) at the point x = a. Illustrate with a sketch.
(c) [3] Use linear approximation to estimate tan−^1 (1.2)
(8) (a) [3] Find all the critical points of the function y = |x| + sin x.
(b) [3] Find the absolute maximum and minimum values of the function y = |x| + sin x on the interval [−π, 2 π].
(c) [2] Does the function y = |x| + sin x have an absolute maximum or absolute minimum on R? (the entire real line). Explain.
(10) [6] Find the length of the longest ladder that can be carried horizontally around the corner of the corridor shown here. (Hint: You are actually looking for the minimum length of something!)
(11) (a) [5] Sketch the parametric curve (x(t), y(t)) where the functions x(t) and y(t) are as below. Indicate on your sketch the points corresponding to the times t = t 1 , t = 0 , t = t 2.
(b) [3] Suppose that a curve given in polar coordinates by the equation r = f (θ) has even symmetry (so if you reflect the curve across the y-axis it looks the same). What can you say about the function f (θ)? (Illustrate your answer by making a sketch of such an f (θ).)
