Calculus I Announcements

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1 Slide 1 Calculus I Announcements Read sections 4.2,4.3,4.4,4.1 and 5.3 Do the homework from sections 4.2,4.3,4.4,4.1 and 5.3 Exam 3 is Thursday, November 12th See inside for a possible exam question.

2 Slide 2 Techniques for Determining Shape To find intervals where f is increasing and decreasing we may use the following procedure: 1. Find the critical numbers of f (and where f is undefined) 2. For each interval between the numbers in part 1 test f to see if it is positive or negative to determine if f is increasing or decreasing on that interval. To find intervals where f is concave up and concave down, we may use the following procedure: 1. Find the values where f is zero or undefined. 2. For each interval between the numbers found in 1, test f to see if it is positive or negative. Try these out on the following function f(x) = x 3 3x

3 Slide 3 First Derivative Test Let c be a critical number of f. If f is increasing for x < c and x near c and f is decreasing for x > c and x near c then f has a local maximum at x = c. In terms of derivatives, we can write this as If f (x) > 0 for x < c and x near c and f (x) < 0 for x > c and x near c then f has a local maximum at x = c. Rewrite these tests for a local minimum.

4 Slide 4 Second Derivative Test Second derivative test for local Maxima and Minima Let c be a number where f (c) = If f (c) > 0 then there is a local minimum at x = c. (The function is concave up near c) 2. If f (c) < 0 then there is a local maximum at x = c. (The function is concave down near c) 3. If f (c) = 0 then the test gives no information.

5 Slide 5 Example For the function f(x) = x 4 4x 3, 1. Find all the critical numbers. 2. Find where the function is increasing and decreasing. 3. Use the first derivative test on each critical number. 4. Use the second derivative test on each critical number. 5. Find where the function is concave up and concave down. 6. Find the inflection points. 7. Make a sketch.

6 Slide 6 iclicker Question Suppose the graph shows f. On which interval(s) is f increasing? A. (0, 1) and (3, 4) B. (1, 3) C. (0, 2) D. (2, 4) E. None of the above

7 Answer to Question Suppose the graph shows f. On which interval(s) is f increasing? A. (0, 1) and (3, 4) B. (1, 3) C. (0, 2) is the correct answer. D. (2, 4) E. None of the above

8 Slide 7 iclicker Question Suppose the graph shows f. On which interval(s) is f concave down? A. (0, 1) and (3, 4) B. (1, 3) C. (0, 2) D. (2, 4) E. None of the above

9 Answer to Question Suppose the graph shows f. On which interval(s) is f concave down? A. (0, 1) and (3, 4) B. (1, 3) is the correct answer. C. (0, 2) D. (2, 4) E. None of the above

10 Slide 8 iclicker Question Suppose the graph shows f. On which interval(s) is f concave down? A. (0, 1) and (3, 4) B. (1, 3) C. (0, 2) D. (2, 4) E. None of the above

11 Answer to Question Suppose the graph shows f. On which interval(s) is f concave down? A. (0, 1) and (3, 4) B. (1, 3) C. (0, 2) D. (2, 4) is the correct answer. E. None of the above

12 Slide 9 Linearization The linearization of a function at a point is just the tangent line at that point. For a function f and a point x = a, the linearization is L(x) = f(a) + f (a)(x a) For the function f(x) = x 1/3 Find the linearization at x = 8 Use the linearization to approximate (8.1) 1/3

13 Slide 10 Linear Approximations This will be a possible exam question on Exam 3. Given the graph of a function and a point on the function, do the following. 1. Sketch the linearization of the function at that point. 2. Sketch the graph of the function and the linearization on zoomed/magnified versions of the same graph.

14 Slide 11 x, y, and dy,dx 1. The quantities x and y refer to properties of the function. (a) x is a number representing a small change in x at a: a + x (b) y is a number representing the change in f for a change of x in x: y = f(a + x) f(a) 2. The differentials dy and dx refer to properties of the tangent line or linearization. (a) dx is called a differential. It is sometimes called an infintesimal as it is infintesimally small. (b) dy is another differential defined by dy = f (a)dx. This is an equation relating infintesimally small entities. 3. The idea that the function behaves locally like its tangent line implies that for small x, y f (a) x The above equation is made exact if we use infintesimals dy and dx instead of y and x. Note: Balanced differentials: In an equation with a dy on one side, there needs to be a dx on the other: dy = f (x)dx

15 Slide 12 iclicker Question Find dy if y = f(x) = sin(2x) A. dy = cos(2x) B. dy = cos(2x)dx C. dy = 2 cos(2x) D. dy = 2 cos(2x)dx E. None of the above

16 Answer to Question Find dy if y = f(x) = sin(2x) A. dy = cos(2x) B. dy = cos(2x)dx C. dy = 2 cos(2x) D. dy = 2 cos(2x)dx is the correct answer. E. None of the above

17 Slide 13 Antiderivatives When we undo a differentiation formula, we call it an anti-derivative. For example: F (x) = x 4 has derivative F (x) = 4x 3... so g(x) = 4x 3 has as an anti-derivative G(x) = x 4 One function can have many anti-derivatives. g(x) = x 3 has G(x) = x 4 /4 as an anti-derivative g(x) = x 3 has G(x) = x 4 /4 + 6 as another anti-derivative g(x) = x 3 has G(x) = x 4 /4 + π as yet another anti-derivative g(x) = x 3 has G(x) = x 4 /4 + C as a general anti-derivative for any constant C

18 Slide 14 Anti-Derivatives: Formulas Here are a few basic formulas, we could include a formula for every function for which you know the derivative. Function General Anti-Derivative f(x) = 0 F (x) = C f(x) = a f(x) = x f(x) = x 2 F (x) = ax + C F (x) = x 2 /2 + C F (x) = x 3 /3 + C f(x) = x n F (x) = xn+1 + C, n + 1 n 1, n 0 { f(x) = 1 x = x 1 F (x) = ln( x ) + C 1 if x < 0 C 2 if x > 0 or F (x) = ln( x ) + C on any interval in the domain of 1/x f(x) = sin(x) f(x) = cos(x) F (x) = cos(x) + C F (x) = sin(x) + C af(x) af (x) + C where F (x) = f(x) f(x) + g(x) F (x) + G(x) + C where F (x) = f(x), G (x) = g(x)

19 Slide 15 iclicker Question Find the general anti-derivative of f(x) = x 2 + 3x + 2 sin(x) A. x 3 /3 + 3x 2 /2 2 cos(x) B. x 3 /3 + 3x 2 /2 2 cos(x) + C C. 2x cos(x) D. 2x cos(x) + C E. None of the above

20 Answer to Question Find the general anti-derivative of f(x) = x 2 + 3x + 2 sin(x) A. x 3 /3 + 3x 2 /2 2 cos(x) B. x 3 /3 + 3x 2 /2 2 cos(x) + C is the correct answer. C. 2x cos(x) D. 2x cos(x) + C E. None of the above

21 Slide 16 Indefinite Integrals The indefinite integral is written as f(x) dx 1. The is an integration sign. 2. The f(x) is called the integrand. 3. The dx is the differential that we learned about earlier. In this context, the dx indicates the variable with respect to which we are anti-differentiating. 4. x is the variable of integration 5. The indefinite integral is a notation for the most general anti-derivative. For example sin(x) dx = cos(x) + C 6. The entire chart of anti-derivatives can be rewritten using the indefinite integral notation (exercise).

22 Slide 17 iclicker 1 Question Find 1 + t dt 2 A. 1 + t 3 /3 + C B t 3 /3 + C C. t 1 t + C D. Use integration by parts E. None of the above

23 Answer to Question Find A. 1 + t 3 /3 + C B t 3 /3 + C t 2 dt C. t 1 t + C D. Use integration by parts E. None of the above is the correct answer. The correct answer is arctan(t) + C

Calculus 221 worksheet

Calculus 221 worksheet Graphing A function has a global maximum at some a in its domain if f(x) f(a) for all other x in the domain of f. Global maxima are sometimes also called absolute maxima. A function