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Unformatted text preview: FINAL EXAM MATH 19 A 6/12/2006 Instructor: Frank Béiuerle, Ph.D. No books, graphing utilities or notes allowed. Show your work. Show
your work. Show your work. Your Name: _____________________________ __ Your TA: ____—._—_...__—_—————_———.——_—_—____—_— Max Your score Problem 1: 30
Problem 2: 10
Problem 3: 10
Problem 4: 30
Problem 5: 10
Problem 6: 10
Problem 7: 20
Problem 8: 15
Problem 9: 15
TOTAL: 150 Good Luck and have a great summer break! 1. (30 points) Compute the derivatives (1—: of the following functions (no need to simplify): (a)y=e (b) y = 111(tan m) (c) y—  l+sin2m Problem continues on next page I Problem continued from previous page 3
(d) y = x In a: (e) y : sinh(1 + arctan m) (f) y = mgr, where m > 0 2. (10 points) (a) State the Mean Value Theorem. (b) Suppose that the distance .3 is a differentiable function of time t for a S t S b. Interpret
the mean value theorem in terms of instantenous and average velocities on [(1,1)]. 3. (10 points) Find the tangent line to the curve
(sin 3:)43’ + (a: + 2)2 2 4y2 at the point (0,1). 4. (30 points) Assume f = 3:3 — 6.22 + 9:13. Answer the following questions. (a) Find and sketch f’(a:). (b) On what intervals is f increasing? (c) On What intervals is f decreasing? (d) Find and classify all local extrema. (e) Find and sketch f”(a:). (f) On What intervals is f concave up? (g) On what intervals is f concave down? Problem continues on next page ! Problem continued from previous page E (h) Find the xcoordinate of all inﬂection points. (i) Sketch the graph of (j) Find the absolute maximum and minimum of f (1:) on the interval [—3, 4]. 5. (10 points) Show that tans: m :2; for a; VERY close to zero. 6. (10 points) Give a graph of a continuous function that satisﬁes the following:
(a) m) = o
(b) f’(rv) = m on [~00,0) ( ) f’(rc) = —2 on (0,1) (d) f”(.’r) > O on (1,2) (e) f’(r) = 0 on (2,3) (1') f”(x) > 0 on (3,00) 0 7. (20 points) Find the area of the largest possible rectangle that can be inscribed in the ellipse $2 y?
_ _=1
9+4 Why does such a. largest area. exist? 8. (15 points) Compute the following limits or explain Why they don’t exist. Justify your steps
by indicating which rule you are using and why it applies. a.) Lime— z——roo :52 b) hm cosa:—1+:L'2 2—40 1:2 — 23 c) lim $1119:
:c—+0+ 9. (15 points) A ladder 20 ft long rests against a vertical wall. If the bottom of the ladder
slides away from the wall at a rate of 1 ft / 5, how fast is the top of the ladder sliding down
the wall when the bottom of the ladder is 12 ft from the wall? ...
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 Fall '06
 Bauerle
 Calculus, Derivative, Mean Value Theorem, Frank Béiuerle

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