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, . . . x_n} such that x_k -x_k-1 = x_j -x_j-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.

lim_n→∞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.