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(j) (k) December 2005 Math 100/180 Final Exam Evaluate
x—r9 f1? — 9
Calculate and completely simplify the derivative of f(a:) 2 In (1000 tan—1(m)). [Note 1 Another notation for tan‘ is arctan]. Find an equation of the line tangent to the graph of 3:3 + 3/3 = 35334 at the point (x,y) :
Find the second derivative of g(m) = sin (621). If we expand sin2(z) in a Maclaurin series, say sin2(x) = Z” cnz", ﬁnd c6. [Hint sin2(x) = %(1— cos(2a:))]. ":0
Find f’(:r) if = (sin Suppose we know that f(1) = f’(1) : 1. Deﬁne g(x) = f(x3). Use a linear approximation
to the function 9 (not a linear approximation to the function f) to estimate g(1.1). Use a suitable linear approximation to the estimate (17)1/4. Give your answer as a fraction
with integer numerator and denominator. What does Taylor’s Inequality give as an upper bound to the error in part (h)? Using Newton’s Method, starting with $1 = 2, ﬁnd the approximation 16;; to the root of
the equation 20m — m3 — 24 = 0. A particle moves in a straight line so that its velocity at time t is v(t) = If its position
at time 9 is 5(9) 2 20, ﬁnd 3(10). ' 2. A circular ferris wheel with radius 10 metres is revolving at the rate of 10 radians per minute. How fast is a passenger on the wheel rising when the passenger is 6 metres higher than the
centre of the wheel and is rising? Include units in your answer. \ passenger ground 3. Wallapak stores her malt beverages outside her house, on her back porch. On a very hot day, she takes one such beverage and places it in her refrigerator, which is constantly kept at
4°C. After 15 minutes, the beverage cooled down to 26 °C, and, after another 15 minutes, it
was at 15°C. How hot was the beverage when it was placed in the refrigerator? Assume the temperature of the beverage obeys Newton’s Law of Cooling. . Find the derivative of f(:r) ; x/l — 2m using the deﬁnition of derivative. No credit will be given
for using differentiation rules, but you may use differentiation rules to check your answer. 5. Let f($) 2 (a) Determine all of the following if they are present: critical numbers, :r—coordinates of local maxima and minima, intervals where f is
increasing or decreasing; m—coordinates of inﬂection points, and intervals where is concave upwards or down-
wards; (iii) equations of any asymptotes (horizontal, vertical, or slant).
(b) Sketch the graph of y = f (at), giving the (at, y) coordinates for all points of interest above.
Draw your sketch on the back of the previous page. 6. At noon, a sailboat is 20 km due south of a freighter. The sailboat is traveling due east at 20
km/h, and the freighter is traveling due south at 40 km/h. (a) When are the two ships closest to one another? Remember to justify your answer.
(b) If the visibility at sea is 10 km, could the people on the two ships ever see each other?
7. (a) A function is deﬁned to equal cos(ax) + b for w 2 0 and 2 — x3 for m < 0, where a and b are constants. It is known that this function is differentiable everywhere. Find all
possible values for a and b. (b) A function g(x) satisﬁes g(0) = 0, 9(1) 2 1, and 9(2) 2 —1. It is known that this function
is twice differentiable everywhere (i.e. g”(:r) exists for all Prove that g’(c) : 0 for
some real number 0. Give complete justiﬁcation, specifying any relevant theorems. The End ...
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