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Math B14-1 Name:
Final Exam I.D.
Fall 1996
Instructions: Write your name and I.D. number above. Show all work on these pages, and make sure that your final answer is clearly shown. No books, calculators, or tables are al- lowed. Check that this exam contains pages 1–6. Good luck, and have a nice holiday!
Circle the name of your instructor: Instructor Section Time Bendel 23 8: DiBenedetto 31 9: DiBenedetto 41 10: Chopp 57 11: Welland 61 12: Chopp 77 1:
Some Useful Formulas
Area of a circle A = πr^2 Surface Area of a sphere A = 4πr^2 Circum. of a circle C = 2πr Volume of a cone V = 13 πr^2 h Area of a trapezoid A = 12 h(b 1 + b 2 ) Volume of a cylinder V = πr^2 h Volume of a sphere V = 43 πr^3 Lateral surface area A = 2πrh of a cylinder
Prob. Possible Score points 1 25 2 25 3 20 4 20 5 20 6 20 7 20 8 15 9 15 10 20 TOTAL 200
- Compute the following derivatives.
(a) (^) dxd (3x^7 − x + 1) (5 pts.)
(b) (^) dxd (x^3 − 1) cos(3x + 1) (5 pts.)
(c) d dx √x^2 −^1 2 x^2 + 1
(5 pts.)
(d) (^) dxd^5
x +
x (5 pts.)
(e) d dx
∫ (^) sin(x) 0
t(t^2 + 1)dt (5 pts.)
- Use implicit differentiation to find the tangent and normal lines to the graph of y^3 − x^3 + 2xy = 2 that pass through the point (1, 2). (20 pts.)
- The function f (x) = x^3 + 4x − 1 has a root in the interval [0, 1] — why? Use x = 1/ 2 as a first estimate for this root and then use Newton’s method to determine a next estimate. (20 pts.)
- The height of a cone is decreasing at the rate of 5 cm/sec while its radius is increasing at 2 cm/sec. When the radius is 4 cm and the height is 5 cm, at what rate is the volume changing; is it increasing or decreasing? ( pts.)
- A right triangle with hypotenuse
3 is revolved about one of its sides to generate a cone. Find the maximum volume of the cone; verify that it is a maximum. (20 pts.)
- Choose an appropriate function and use linear approximation at an appro- priate value to estimate
(15pts.)
Sketch the graphs of f (x) = −x + 1 and g(x) = x^2 − x and find the area of the region above the graph of g(x) and below that of f (x). (20 pts.)
