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Unformatted text preview: “ FINAL EXAM
MATH 19 A 07/25/2003 Instructor: Frank B'ziuerle, Ph.D. Your Name: _— —————__——.—.————_.__._______——__—.—__—_—. Max Your score Problem 1: 30
Problem 2: 20
Problem 3: 10 Problem 4: 20
Problem 5: 10
Problem 6: 10 Problem 7: 20
Problem 8: 20
Problem 9:10 Problem 10: 20
Problem 11 (EC): 10
TOTAL: 170 Good Luck ! 1. (30 points) Compute the derivatives 3:: of the following functions (no need to simplify): (a) y = sin2 a: (b) y = ln(cosh 2:) Problem continues on next page I Problem continued from previous page !
(d) y = 2:3 ln .7: (e) y = tan(1 + sin 3) (f) y = main: (Hint: Use logarithmic differentiation.) 2. (20 points) (a) State Rolle’s theorem or the mean value theorem. (b) Show that m5 + a: + 1 = 0 has exactly one real root. 3. (10 points) Find the equation of the tangent ﬁne to the curve
3:43; + 5”"1 = 23: at the point (1, 1). 4. (20 points) Use the given graph of f’(:1:) (the DERIVATIVE of f (3)) to answer the following
questions about f(:1:) on the interval [0,6]. (a) On what intervals is f increasing? (b) On what intervals is f decreasing? (c) Find the :ccoordinate of all local maxima ((1) Find the :c—coordinate of all local minima. (e) On what intervals is f concave up? (f) 011 what intervals is f concave down? (g) Find the xcoordinate of all inﬂection points 5. (10 points) Show that sin a: 9:3 :2: for a: VERY close to zero. 6. (10 points) (a) Give an example (graph and equation) of a function f(:z) where f (0) = f’ (0] = f"(0] = 0
but a: = 0 is not an inﬂection point of f (x) (b) Suppose we know that f”(:c) < 0 for 1 _<_ a: S 3, f’(2) = 0, and f(2) = 3. Give a rough
sketch of f (as) of the graph of f locally around .7: = 2. 7. (20 points) Assume you have 400ft of fencing available. You want to build a rectangular
enclosure with a. center divider alongside a. long, straight building. What are the dimensions
that yield the maximal area of the enclosure? What is this maximal area ? 8. (20 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)1im< 5) r——+1 1—33 1—3 b) lim ___E “i“ z—+0 a: + 0032: c) lim —— :—mo 3; 9. (10 points) Boyle’s Law states that when a sample of gas is compressed at a constant
temperature, the pressure P and volume V satisfy the equation PV=C Where C is a constant. Suppose that at a certain instant the volume is 600 cm3, the pressure
is 150 kPa, and the pressure is increasing at a rate of 20 kPa/rnin. At what rate is the
volume increasing at this instant? 10. (20 points) Let f(:c) = $4 + 4.13. Find the following: (a) m—intercepts: (b) Critical points: (c) Local max/min. Justify your answers. ((1) Inﬂection points: (e) Sketch the graph of f. Label all points carefully. 11. (Extra Credit, 10 points) Assume that f(:1:) is differentiable at :r = a. Compute the
following limit: Hm ms) — f(a) 1—“; $2 __ a2 Justify your steps. 10 ...
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 Fall '06
 Bauerle
 Calculus

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