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This is the Exam of Calculus which includes Start Biking, Square Inch, Some Points, Solution, Slope, Slope Fields etc. Key important points are: Simplify Tan, Circle, Notes, Evaluate, Simplify, Arctan, Function, Limit Definition, Derivative, Certain Moment
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Math 105D - Final Exam - December 12, 2006
Instructions: Show all of your work and circle your final answers. Calculators are allowed, but notes and books are not.
x ln x
(b) (5 points) Simplify tan(arcsin 3x).
(c) (6 points) Find
d dx
x(arctan 2x)(ln x) + cos (ex
2
3 x
. Use the limit definition of the derivative to find f ′(x).
0
(x^2 + 1) dx.
(a) Use the Fundamental Theorem of Calculus to evaluate this integral exactly.
(b) Approximate the integral using right endpoints with three subintervals.
(c) Write your answer to part (c) in summation notation (i.e., using
(Problem 8 continued - still working with
0
(x^2 + 1) dx.)
(d) Use summation notation to express the approximation of the integral with right endpoints on n subintervals.
(e) Would a Riemann sum with left endpoints on 50 subintervals give an underestimate, an overes- timate, or can we not be sure? Explain. (A picture may be helpful. Note: You do not have to calculate L 50 .)
