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The past exam paper of Calculus, key points are: Graph, Region Bounded, Exact Area, Region, Region First, Sketch, Integral Representing, Origin, Definite, Arc Length
Typology: Exams
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EXAM I - SEPTEMBER 30, 2005
Instruction: Read each question carefully. Explain ALL your work and give reasons to support your answers. Advice: DON’T spend too much time on a single problem.
Problems Maximum Score Your Score
1
2 EXAM I - SEPTEMBER 30, 2005
1.(15 pts.)(a) Consider the region bounded by the graph of y = (x−1)^3 +1, the line y = 1 and the graph of y = x^2. Find the exact area of the region. [Make a sketch of the region first.]
(5 pts.)(b) Write (DO NOT evaluate) a definite integral representing the arc-length of the path given by y = (x − 1)^3 + 1 from the origin (0,0) to the point (1,1).
4 EXAM I - SEPTEMBER 30, 2005
0 f^ (x)^ dx.
(8 pts.)(b) Recall that the error committed by using the Right Hand Sum Rn is less than or equal to K^1 ·( 2 bn− a)^2 where |f ′(x)| ≤ K 1 for some constant K 1. Suppose that |f ′(x)| ≤ 6 for 0 ≤ x ≤ 2. How large do you require n to be in order to guarantee that |Rn −
0 f^ (x)^ dx| ≤^0 .001?
MATH106A CALCULUS II - PROF. P. WONG 5 4.(10 pts.)(a) Evaluate the indefinite integral ∫ (^) cos( 1 t ) t^2 dt
(10 pts.)(b) Let F (x) =
∫ (^) x 1
sin(t^2 ) dt.
On the interval [0, √π], where is F (x) concave up? Justify your answer.
