Reference no: EM131122422
MAT 137Y: Calculus Problem Set-
1. Below is the graph of the function f:
We define a new function by F(x) = 0∫x f(t)dt. Answer the following questions about F and justify your answers.
(a) Is F(6) positive or negative?
(b) At which point or points does F have a local minimum?
(c) Where is the function F concave up?
(d) On the domain [-1, 5], at which point or points does F have an inflection point?
(e) On the domain [-1, 5], at which point or points does F have a global minimum?
(f) Find two values of x such that F(x) = 0.
2. Recall, a regular partition is a partition {x0, . . . xn} such that xk -xk-1 = xj -xj-1 for every k and j. Evaluate -3∫2(x+1)2dx using a Riemann sum with a regular partition. Use a partition with n points and take the limit as n → ∞.
3. Approximate 0∫1/2 sin(2πx)dx using a finite Riemann sum with the partition {0, 1/12, 1/8, 1/4, 3/8, 1/2}.
4. Evaluate the following limit. Hint: it is a Riemann sum using right endpoints and you'll want to use The Fundamental Theorem of calculus. You are not allowed to sum directly using a formula.
limn→∞k=1∑n729k5/n6 (1)
5. Find the area bounded between x - y2 + 1 = 0 and x - y - 1 = 0.
6. Prove The Mean Value Theorem for Integrals: Let f : [a, b] → R be a continuous function. Then there exists c ∈ [a, b] such that a∫b f(t)dt = f(c)(b - a).
7. Use the chain rule to show to evaluate d/dx x2-1∫sin(x) cos(t)dt.
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