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A final exam for a university math 105 course, including calculus problems, limits, derivatives, taylor polynomials, newton's method, optimization, and approximations. The exam includes 8 questions with multiple parts, covering various topics in calculus.
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SECTION: (circle one) 11:00-11:55 12:05-1:
Math 105 - Final Exam - December 9, 2005
Instructions: Show all of your work and circle your final answers. Calculators are allowed, but notes and books are not.
(a)
1
(2x + 3) dx.
(b)
d dx
(∫ (^) x
3
sin (t^2 ) dt
(c)
0
(e^2 r^ −
r) dr.
(d) If y = e^3 x+4^ cos
x
, find y′.
(e) lim x→∞
5 + sin x 2 x^
(f)
− 3
9 − u^2 du.
(g) lim x→ 8
x − 8 √ (^3) x − 2.
f (xn) f ′(xn)
(a) The function f (x) = x^3 − 6 x^2 + 7x + 3 has a root in the interval [1, 3]. Using an initial guess of x 0 = 2, give the first three approximations for this root. (Give your answers to 8 decimal places.)
(b) Verify that an initial guess of x 0 = 1 will not approximate the root of f (x) in the interval [1, 3]. Explain, using a graph if necessary, why Newton’s method fails to find the correct root with this initial guess.
(a) Using a midpoint sum with 3 rectangles, calculate an estimate of
0
x x^3 + 1
dx.
(b) For the same integral, use Σ-notation to express the right sum approximation with 6 rectangles. (You do not need to calculate the exact value.)
