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.