代做Math3b Week 2 :Fundamental Theorem of Calculus代做留学生SQL语言程序

Week 2 :Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus, Part 1.

If f is continuous on [a, b], define

Then g is continuous on [a, b], differentiable on (a, b), and

g′(x) = f(x).

The Fundamental Theorem of Calculus, Part 2.

If f is continuous on [a, b] and F is any antiderivative of f on [a, b] (that is, F′ = f), then

Problem 2. Let

where f is the function whose graph is shown below.

(a) Evaluate g(x) for x = 0, 1, 2, 3, 4, 5, 6. (b) Estimate g(7).

(c) Where does g have a maximum value? A minimum value? (d) Sketch a rough graph of g.

6. The graph of a function f is shown. Let g be the function that represents the area under the graph of f between 0 and x.

(a) Use geometry to find a formula for g(x).

(b) Verify that g is an antiderivative of f and explain how this confirms Part 1 of the Fundamental Theorem of Calculus for the function f.

Problem 10. Use Part 1 of the FTC to find the derivative of :

Problem 16. Use Part 1 of the FTC to find the derivative:

Problem 32. Evaluate the integral:

Problem 62. On 0 ≤ x ≤ π/3,

(a) Use a graph to give a rough estimate of the area under the curve y = sec2x.

(b) Then find the exact area.