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Integral Calculus

Integral Calculus. AP Calculus AB Mr. Reed. Linear Approximations & Differentials.

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Integral Calculus

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  1. Integral Calculus AP Calculus AB Mr. Reed

  2. Linear Approximations & Differentials • By now we have seen that if a function is differentiable at a point, we can find a linear function that exactly mimics this function at that point. In this section we will study the further use of linear approximations and learn about differentials. • Reproduce sketch showing f(x), l(x), Δx, Δy, dx, dy

  3. Local Linearity and Linearization of a function • If f is differentiable at x = c, then the linear function, l(x), containing (c, f(c)) and having a slope of f’(c) (the tangent line) is a close approximation to the graph of f for values of x close to c. • Linearizing a function f means approximating the function for values of x close to c using the linear function:

  4. Error in linear approximation • To find the error in using a line to approximate a function’s value use: Error = f(x) – l(x) • Examples: #2, p.194 (Foerster)

  5. Finding differentials • One method for finding differentials uses a linear approximation equation. From the linear equation, study the ratio of f’(c)(x-c) to (x-c). • If dx and f’(x) are known, then dy can be found by multiplying f’(x) and dx. • Examples: #10, #12 p.196 (Foerster)

  6. Going backwards – finding the antiderivative • Examples: #28, #30 p.196 (Foerster) • HW  p.194-196: Q1-Q10, 1, 3, 9-17 (odd), 27-35 (odd)

  7. Indefinite Integration and Antidifferentiation • Let g(x) is the antiderivative f(x), the stretched out S is the integral sign, f(x) is called the integrand, dx is the differential, and the whole expression is called the integral.

  8. Two Properties of Indefinite Integrals • Integral of a constant times a function: • Integral of a sum of two functions:

  9. Integral of a Power Function • For any constant n ≠-1 and any differentiable function u,

  10. Integral of Exponential Functions • If u is a differentiable function, then • If base other than e, simply divide by a factor of ln(b), where b is the base of the exponential function

  11. Examples integrating using ‘u-substitution’ • Examples from p.202 (Foerster): 4, 8, 10, 12, 14, 18, 20, 22, 26, 30 • HW ==> p.202-203: Q1-Q10, 1-31(odd), 34 (need trapezoidal rule on calculator)

  12. Riemann Sums • A Riemann sum is an approximation of the definite integral of f(x) with respect to x on the interval [a,b] using rectangles to estimate areas.

  13. Left, Right, Midpoint Riemann Sums • Left  uses left side of rectangle to get height (Ln) • Right  uses right side of rectangle to get height (Rn) • Midpoint  uses middle of rectangle to get height (Mn) • See p. 206 diagrams (Foerster)

  14. Lower & Upper Riemann Sums • Lower  Each rectangle is “under” curve (Ln) • Upper  Each rectangle is “above” curve (Un) • See p.207 diagrams (Foerster)

  15. Definite Integral & Integrability • If the lower sums, Ln and the upper sums, Un, for a function f on the interval [a,b] approach the same limit as Δx approaches zero (or as n approaches infinity in the case of equal-width subintervals), then f is integrable on [a,b]. This common limit is defined to be the definite integral of f(x) with respect to x from x = a to x = b. The numbers a and b are called the lower and upper limits of integration, respectively. Algebraically, provided the two limits exist and are equal.

  16. The Fundamental Theorem of Calculus • Recall that: If we want to evaluate the definite integral, we can do so by evaluating the limit. (show example) • However, by using the Mean Value Theorem, it can be shown that: where g(x) is the antiderivative of f(x).

  17. Two uses of FTC • Used to evaluate definite integrals • Used to evaluate antiderivative at upper or lower bounds of integration.

  18. Examples Using FTC & Properties of Definite Integrals • P.232 (Foerster)  2, 6, 8, 14, 18, 24

  19. Properties of Definite Integrals • See p.231 for table highlighting the following properties of definite integrals: • Positive and Negative Integrands • Reversal of Limits of Integration • Sum of Integrals with Same Integrand • Integrals Between Symmetric Limits • Integral of a Sum and of a Constant Times a Function • Upper Bounds for Integrals

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