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Unformatted text preview: December 2004 Math 100/180 Final Exam 1. Short  Answer Questions. $2 — 1
(a) Evaluate i113 m. , 1162 ~ 1
(b) Evaluate xii—{£10 m. (C) Calculate the derivative of f(m) : sin—10:2). [Notez Another notation for sin"1
is arcsin] (d) The curve y : 306””C has one inﬂection point. Find the 1' coordinate of this
point. {E3 + 1132 m2 + 1 ' (f) Find an equation of the tangent line at (0,1) to the curve 7:3 + y3 2 sinx + 1. (e) Find an equation of the slant asymptote to the curve y : (g) Find a number 250 between 0 and g such that the tangent line to the curve y = cosy; at :3 2 1'0 is parallel to the line y : —§.
1 . . dy
(h) If y z m “I, ﬁnd the derivative d—
an (i) Using a linear approximation, estimate f(4.1), given that f(4) = 3 and f’($) =
\/1L‘ + 9.
(j) The function f (m) is deﬁned by 1'3 ifx<1
ﬂat): .
A$+B iwaI Where A and B are constants. For what values of A and B is f(:1:) differentiable
for all as? (k) Starting with the initial guess 1:0 2 1, Newton’s method is used to approximate
a solution to the equation (tan‘1 :5) — m2 = 0. Find the next approximation, :51,
and simplify your answer as much as you can. [Notez Another notation for tan‘1
is arctan] 1035—1. What is the maximum (1) A certain function has second derivative equal to 6
error that can occur when the linear approximation to the curve at as = 0 is used to approximate f (0.1) 7 ﬂ“— 2. 3. 4. (m) Determine the ﬁrst four nonzero terms in the Taylor series for f (m) = cos(ac) + sin(2$) at x = 0 (i.e. the Maclaurin series). $2 (11) The function f(m) = 67 has Maclaurin series co + 012: + czar2 + ~ .  . Determine the value of C4. Full — Solution Problems. In questions 2— 6, justify your answers and show all
your work. A turkey is put into an oven that has a constant temperature of 200°C. A ther
mometer embedded in the turkey registers its temperature. When the turkey is put
into the oven, the thermometer reads 20°C, and 30 minutes later it reads 30°C. the
turkey will be ready to eat when the thermometer reads 80°C. How many minutes
after being put into the oven will the turkey be ready to eat? Assume that the
turkey’s temperature satisﬁes Newton’s law of cooling. For the function 2 f($‘) = 332—1 determine all of the following if they are present: (i) Critical points, local maxima
and minima, intervalsbvhere f (at) is increasing or decreasing; (ii) inﬂection points
and intervals where f (:13) is concave upward or downward; (iii) asymptotes (hori—
zontal, vertical, slant). Sketch the graph of y = f (:13), giving the (:L',y) coordinates
for all the points of interest above. A streetlight is on the top of a vertical pole that is 8 m high. A ball is dropped from
a helicopter and falls vertically, landing 3m away from the base of the streetlight
pole. When the ball is below the level of the streetlight7 the light casts a shadow
of the ball on the ground. When the ball is 4 m above the ground, it is falling at a rate of 20 m/ see. (a) When the ball is 4 m above the ground, how fast is the shadow moving? (Assume
that the ground is ﬂat). (b) When the ball is 4 m above the ground, how fast is the distance between the
ball and its shadow Changing? 5. A swimming pool is to be constructed in the shape depicted below. It has a shallow end of depth 1 In and a deep end of depth 2 m, and the shallow end is twice as long
as the deep end. The total volume of the pool is 800 m3. What should the width
of the pool be so that the total area of the seven surfaces that are in contact with water when the pool is completely ﬁlled with water (Le. all surfaces except the open
top surface of the pool) is minimized? Figure 1: Swimming pool. 6. (a) Find the derivative of
1 1—m2 using the deﬁnition of the derivative. No marks will be given for the use of differ
entiation rules. (b) Suppose f (m) is a function that is differentiable for all 1‘. Let g(x) be the new function deﬁned by g(:1c) = f(:1:) + f(1 — 2:2). Prove that g’(c) = 0 for some positive
real number 0. ...
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 Fall '08
 LAMB
 Math

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