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Unformatted text preview: Math 31L.01  Test 2 READ INSTRUCTIONS BELOW BEFORE STARTING THE EXAM 0 Do not open this test booklet until you are directed to do so. 0 You will have 105 minutes to complete the exam. If you ﬁnish early, go back and check
your work. 0 Throughout the exam1 show your work so that your reasoning is clear. Otherwise no credit
will be given. Circle your answers. a Do not spend too much time on any one problem. Read them all through ﬁrst and attack
them in an order that allows you to make the most progress. I have adhered to the Duke Community Standard in completing this assignment.
Name: Signed: 1. (10 pts) Two of the following differential equations are easy to solve. Pick those two and
solve them. In each case the initial condition is y(0) = 1. aﬂ=3—1.5y ioiﬂ=3—1.5y2 cﬂ=3—1.5t2
dt d: dt 2. (10 pts) A squarebottomed box With a top must be designed to have a fixed volume of 12
fig. What dimensions minimize the surface area of the box? Be sure to Show all of your
work, and include justiﬁcation that your answer is actually a. minimum. 3. (5 pts) The derivative of which of the following functions is a good approximation of the
graph of the derivative of sin x? Circle ONE answer: Iimhﬁo sigh — sin 3 — cos 1' arcsin a: arceos m
sin{:v+.001)v—sina: cos(w+.001)~cos:z:  sing2 +h)—sin 12
.001 .001 hth!) h 4. (8 points) Consider the curve given by m3 + y3 — myg = 5, (3.) Using implicit differentiation, ﬁnd g. (b) Find the equation of the tangent line to the curve through the point (2, —1). 5. (12 pts) Derivatives of Inverse Functions (a) (3 pts) Give a. simple counterexample to show that: _1__
(f )(x) 75 m) (b) (4 pts) Assume f(:n) is an invertible, differentiable function. Find (f‘1)’ (5) using the
following information: (c) (5 pts) In class, we used the Chain Rule to show that %(arcsin 3:) = Show
how we did that. 6. (10 pts) Suppose a. small bug is moving along a straight line and her position at time t is
given by 3(t). The bug has an acceleration given by —4t cm/secg, Where t is the number
of seconds since she began moving. Her initial position is at s = 10 cm and her velocity at
time t = 1 is 24 cm/sec. What is her location on the line at time t = 3? 7. (5 points) Find limmnq $131, if it exists; justify your answer. 8. (10 pts) A train is moving along a straight track, moving in the direction shown in the
ﬁgure below. A video camera, 0.5 km away from the track, is focused on the train. When
the train is one kilometer from the camera, the camera is rotating at 0.43 radians/minute.
How fast is the train moving at that moment? Be sure to include proper units. 'q. ..v .. _.A 0.5 d _
9. (8 pts) Suppose Ema — :ccosac + (a) Show that 31(33) = a: sine: is a solution to the above diﬁ‘erential equation. (b) For what values of the constant c is y(:c) = msinm + c a solution to the differential
equation? 10. (Bonus: 2 pts) L’Hopital became a mathematician because:
(a) He wanted to impress the lovely Evita Nercelene, who was herself an amatuer math
ematician.
(b) His poor eye sight prevented him from pursuing his planned military career.
(0) He failed all his literature courses miserably. (6.) He was to poor to just be a gentleman like his father. 11. (12 pts) Consider the function f (as) s (1326'? where a > O is some positive constant. (a) Find the location of all local maxima and minima; justify your answers. (13) Use L’Hopital’s rule to ﬁnd limmnm ﬁx); also ﬁnd limz.._Do f (c) Using you answers to the ﬁrst two parts, give a rough sketch of ﬁx) when a = 1. (d) Does f have a global maximum? If so, identify Does f have a. global mini—
mum? If so, identify it. 12. (10 pts) Assume ﬁx) is at twice differentiable function. For each of the following, deter—
mine if the statement is true or false. If it is false, give or sketch an example of a function
that contradicts the statement. (a) If f(m) has both a local max and a local min at the point :3 = a, then ﬂat) is a
constant function. (b) On a closed interval, f (:13) has a global max and a global min. (c) On an open interval, f cannot have both a global max and a global min. (d) On an open interval, the local extrema of f must occur at critical points. (e) If f’(a,) = D and f”(a) = 0, then f(m) has an inﬂection point at a: = a. ...
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This note was uploaded on 04/28/2009 for the course MATH 31l taught by Professor Staff during the Spring '08 term at Duke.
 Spring '08
 Staff
 Calculus

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