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Example

Example. Ex. Find Sol. So. Example. Ex. Find (1) (2) (3) Sol. (1) (2) (3). Question. Find Sol. Use L’Hospital’s rule. Question. Find Sol. Optimization problems. Optimization : minimize costs and/or maximize profits

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Example

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  1. Example Ex. Find Sol. So

  2. Example Ex. Find (1) (2) (3) Sol. (1) (2) (3)

  3. Question Find Sol. Use L’Hospital’s rule.

  4. Question Find Sol.

  5. Optimization problems • Optimization: minimize costs and/or maximize profits • Steps in solving optimization problems: first understand the problem and formulate the cost function, then find the global minimum/maximum using the closed interval method.

  6. Example Ex. Find the area of the largest rectangle that can be inscribed in a semicircle of radius r. Sol. Set up the coordinate system. The semicircle has the equation Let (x,y) be the vertex lying in the first quadrant. Then the rectangle has length 2x and width y, so its area is A=2xy. Since we can eliminate y: thus The domain is [0,r]. Use the closed interval method, A(x) has maximum value

  7. Newton’s method • Find a root of f(x)=0 • Idea: successively replace f(x) by its linear approximation. Given an initial guess of a root, say, approximate f(x) by the linear approximation of f(x) at Use the root of as the approximate root of f(x)=0, and denote it as That is, so Repeat this process, we obtain a sequence with recurrence relationship:

  8. Newton’s method • Under appropriate conditions, the sequence generated from Newton’s method is convergent to the root of f(x)=0. • Ex. Without using the operation of taking roots, find correct to four decimal places. • Sol. is a root of Since Newton’s method gives Choosing an arbitrary say we have Since and agree to 4 decimal places, we conclude that correct to four decimal places.

  9. Antiderivatives • Definition A function Fis called an antiderivative of f on an interval I if for all x in I. • For example, is an antiderivative of • Question: given f(x), is the antiderivative of f unique? • Theorem If F is an antiderivative of f on an interval I, then the most general antiderivative of f on I is F(x)+C where C is an arbitrary constant.

  10. Example • Ex. Find the most general antiderivative of : (1) (2) (3) • Sol. (1) (2) (3)

  11. Example • Ex. Find all functions g such that • Sol. By the sum rule of derivative, we can find the antiderivative for each term and add together.

  12. Example • Ex. Find all functions g such that • Sol. Write the function into the sum of the functions, for which we can find antiderivative.

  13. Example • Ex. Find g if g(1)=0 and • Sol.

  14. Direction fields • The geometry of antiderivatives can be described by a direction field: given f, to draw the graph of F, at an arbitrary point x, the tangent line has slope f(x) • Ex. If sketch the graph of the antiderivative F that satisfies the initial condition F(-1)=0.

  15. Homework 11 • Section 4.7: 17, 22 • Section 4.10: 27, 29, 32 • Review exercises (P362): 13, 14, 51

  16. Question for midterm review Suppose that f(x) is defined for all and that for any real number x,y, where Suppose also that Find Sol.

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