Test2 - Math 31L.01 - Test 2 READ INSTRUCTIONS BELOW BEFORE...

Info iconThis preview shows pages 1–9. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 8
Background image of page 9
This is the end of the preview. Sign up to access the rest of the document.

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 square-bottomed 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. ...
View Full Document

This note was uploaded on 04/28/2009 for the course MATH 31l taught by Professor Staff during the Spring '08 term at Duke.

Page1 / 9

Test2 - Math 31L.01 - Test 2 READ INSTRUCTIONS BELOW BEFORE...

This preview shows document pages 1 - 9. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online