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Linear Approximations and Differentials in Calculus

Learn about linear approximations and differentials in calculus, understanding local linearity, error in linear approximation, finding differentials, antiderivatives, indefinite integration, Riemann sums, definite integral, integrability, and the Fundamental Theorem of Calculus.

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Linear Approximations and Differentials in 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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