Ch. 5 Review: Integrals

1. Ch. 5 Review: Integrals AP Calculus

2. 5.2: The Differential dy 5.2: Linear Approximation 5.3: Indefinite Integrals 5.4: Riemann Sums (Definite Integrals) 5.5: Mean Value Theorem/Rolle’s Theorem Ch. 5 Test Topics

3. dx & dy: change in x and y for tangent (derivative) Tangent line The Differential dy

4. Find the differential dy: y = dy = (6x – 4) dx

5. Write the equation of the line that best fits at x = 2. Then find dx, and dy if f(2.01) is approximated. Equation: dx dy Linear Approximation

6. Write the equation of the line that best fits at x = 2. Then find dx, and dy if f(2.01) is approximated. Point of tangency: f(2) = -2 Slope of tangent (deriv): y’ = 6x – 7 when x = 2  5 Sub into pt-slope equation: y – y + 2 = 5(x – 2)  y = 5x – 12 If x = 2.01, y = -1.95 : Function change in y: f(2.01) – f(2) = .0503 dx: Tangent line change in x -- 2.01 – 2 = .01 dy: Tangent line change in y for x = 2 to 2.01: -1.95 - -2 = .05 or dy = f’(x) dx at x = 2  (6(2) – 7)(.01) = .05 Linear Approximation

7. If a function is continuous and differentiable on the interval [a, b], then there is at least one point x = c at which the slope of the tangent equals the slope of the secant connecting f(a) and f(b) Mean Value Theorem

8. If a function f is: • Differentiable for all values of x in the open interval (a, b) and • Continuous for all values of x in the closed interval [a, b] Then there is at least one number x = c in (a, b) such that f’(c) = Mean Value Theorem (MVT)

9. If a function is differentiable and continuous on the interval [a, b], and f(a) = f(b) = 0, then there is at least one value x = c such that f’(c) = 0. Rolle’s Theorem

10. Remember – Function must be CONTINUOUS and DIFFERENTIABLE on interval! Otherwise, conclusion of MVT may not be met. Mean Value Theorem

11. Integrals Self-Quiz

12. Integrals Self-Quiz

13. Integrals Self-Quiz

14. Integrals Self-Quiz

15. R Problems, pg. 260: R1 –R5 ab