What makes f '(x) undefined? (set the denominator = 0)

Transcription

1 Chapter 3A Review 1. Find all critical numbers for the function ** Critical numbers find the first derivative and then find what makes f '(x) = 0 or undefined Q: What is the domain of this function (especially since I see a radical we can't have any imaginary numbers!!) ** Use the product rule to find the derivative f(x) g '(x) + g(x) f '(x) Drop the negative exponent to the bottom of the first term Write this as 1 term that means we have to find an LCD and multiply the second term (top & bottom) by what's missing What makes f '(x) = 0 (set the numerator = 0) What makes f '(x) undefined? (set the denominator = 0) 1

2 2. Find all critical numbers for the function: Set f '(x) = 0 and solve take out a GCF 3. Use a graphing utility to graph the function. Use the graph to find the value of the derivative (if it exists) at the extremum. ** Q: What is the value of the derivative at a max or min (if it exists)? A relative min exists at x = 2, Point (2, 0) but f(x) is not differentiable at x = 2, there is a corner in the graph. f '(x) DNE 4. Determine from the graph whether f possesses extrema on the interval (a, b) 2

3 5. Decide whether Rolle's Theorem can be applied to the function on the interval [0, 2]. If Rolle's Theorem can be applied, find all value(s) of x in the interval such that f '(c) = 0. If Rolle's Theorem cannot be applied, state why. ** Recall: In order to apply Rolle's Theorem, the following must be true 1. The function must be continuous on [a, b] 2. The function must be differentiable on (a, b) 3. f(a) = f(b) Since the function is a quadratic polynomial, it is continuous on [a, b] and differentiable on (a, b) Since f(0) does not equal f(10), Rolle's Theorem cannot be applied. 6. Use a graphing utility to graph the function. Use the graph to determine on which of the following intervals Rolle's Theorem applies. 3

4 7. Given the function, find all c in the interval (2, 7) such that So we need to find a value (c) where the derivative of f(x) equals 1 Now set f '(x) = 1 and solve Cross multiply to solve for x BUT the interval is (2, 7) are both values included in the interval? 8. MISTAKE ON PROBLEM: Determine whether the Mean Value Theorem applies to the function on the stated interval. If the Mean Value Theorem applies, find all value(s) of c in the interval such that: If the Mean Value Theorem does not apply, state why. f(x) is continuous on [ 3, 2/3] and differentiable on ( 3, 2/3) Set f '(x) = 1/2 4

5 9. State why the Mean Value Theorem does not apply to the function on the interval [ 3, 0] A: B: f is not continuous at x = 1 C: Both a and b D: f is not defined at x = 3 and x = 0 E: None of these 10. Find all open intervals on which the function is decreasing. Do the quotient rule to find the derivative Increasing or decreasing question? Think first derivative and critical points!! What is the critical point(s)? Interval Test Value x = 1 x = 1 Sign f '(x) f ' (x) < 0 f '(x) > 0 Conclusion Decreasing Increasing 5

6 11. Find all open intervals on which the function is decreasing Critical Numbers? Interval Test Value x = 3 x = 0 x = 2 Sign f '(x) f '(x) < 0 f '(x) < 0 f '(x) < 0 Conclusion Decreasing Decreasing Decreasing 12. Which of the following statements is true of A: f is increasing on (, 5) B: f is decreasing on (5, 7) C: f is increasing on (5, 7) D: f is decreasing on (6, ) E: None of these 6

7 13. Use the graph to identify the open intervals on which the function is increasing or decreasing 14. Find the values of x that give relative extrema for the function A: Relative maxima: x = 1, 1; relative minimum: x = 0 B: Relative maximum: x = 0; relative minima: x = 1, 1 C: Relative maximum: x = 1; relative minimum: x = 1 D: Relative maximum: x = 0; relative minimum: x = E: None of these Critical points: Interval Test Value x = 2 x = 1/2 x = 1/2 x = 2 Sign f '(x) f '(x) > 0 f '(x) < 0 f '(x) < 0 f '(x) > 0 Conclusion Increasing Decreasing Decreasing Increasing 7

8 15. Find all relative extrema of the function Max & Min problem? Think first derivative, critical points and increasing/decreasing ** Special note you can't use your calculator for these problems (other than checking your answer)!!! You must do the work!! Using the calculator to find the max/min is not an acceptable use of the calculator on the AP Exam Let's factor out a GCF GCF: (x + 4) 2 Critical Points: We can cancel the common factors Interval Test Value x = 5 x = 0 x = 3 Sign f '(x) f '(x) > 0 f '(x) > 0 f '(x) < 0 Conclusion Increasing Increasing Decreasing 8

9 16. Given than the function has a relative maximum at x = 6, choose the correct statement A: f ' is negative on the interval (6, ) B: f ' is positive on the interval (, ) C: f ' is positive on the interval (6, ) D: f ' is negative on the interval (, 6) E: None of these 17. Let f(x) = (x + 2) 3 4. The point ( 2, 4) is A: A critical point but not an extremum B: An absolute maximum C: Not a critical point D: An absolute minimum E: None of these Critical Point: Interval Test Value x = 3 x = 0 Sign f '(x) f '(x) > 0 f '(x) > 0 Conclusion Increasing Increasing 9

10 18. Show that f has no critical numbers Recall: Critical numbers must be defined in f(x) and they are values that make f '(x) = 0 or undefined What makes f '(x) = 0? What makes f '(x) undefined? Are all of these values defined in f(x)? 19. Find all intervals on which the graph of the function is concave upward: ** Concavity? Think second derivative and inflection points Find the inflection points by finding where f ''(x) = 0 or where f ''(x) is undefined Inflection Points: Interval Test Value x = 1 x = 1 Sign f ''(x) f ''(x) > 0 f ''(x) > 0 Conclusion Concave up Concave up 10

11 20. Choose the correct statement for the given function A: The graph of f is concave upward on the interval (0, ) B: The graph of f is concave upward on the interval (, ) C: The graph of f is concave upward on the interval (, 0) D: The graph of f is concave downward on the interval (, 0) E: None of these Inflection Points: Interval Test Value x = 1 x = 1 Sign f ''(x) f ''(x) > 0 f '(x) < 0 Conclusion Concave up Concave down 21. Find all points of inflection of the graph of the function: Inflection Points: 11

12 22. Find all points of inflection of the graph of the function: 23. Let f ''(x) = 4x 3 2x and let f(x) have critical numbers 1, 0 and 1. Use the Second Derivative Test to determine which critical numbers, if any, give a relative maximum. Point x = 1 x = 0 x = 1 Sign f ''(x) f ''(x) < 0 f ''(x) = 0 f ''(x) > 0 Conclusion Relative Max Test Fails Relative Min 24. Let f(x) be a polynomial function such that f( 2) = 5, f '( 2) = 0, and f ''( 2) = 3. The point ( 2, 5) is a on the graph of f. A: Point of Inflection B: Intercept C: Relative maximum D: Relative minimum E: None of these 12

13 25. Let f(x) be a polynomial function such that f(4) = 1, f '(4) = 2, and f ''(4) = 0. If x < 4, then f ""(x) < 0 and is x > 4, then f ''(x) > 0. The point (4, 1) is a of the graph of f. A: Critical Number B: Relative Minimum C: Point of Inflection D: Relative Maximum E: None of these 26. Give the sign of the second derivative of f at the indicated point 13

14 27. Which statement is not true of the graph of A: f has a point of inflection at (3, 0) B: f has an intercept at (3, 0) C: f has a relative minimum at (3, 0) D: f has a relative maximum at (1/3, 256/27) E: None of these #1: Intercepts: set the factors = 0 and solve for x #2. First derivative test (for max/min values) Critical Numbers: x = 0 x = 1 x = 4 f ' > 0 f ' < 0 f ' > 0 Increasing Decreasing Increasing #3. Find the points of inflection Point of Inflection: 14

15 28. Find the horizontal asymptote Divide every term by the term in the denominator in this case by x ** Remember! and the limit of a constant (a number) is that number 29. Find all horizontal asymptote for the function That radical in the bottom tells me I will probably have two asymptotes one from and one from. Verify with a graph if you are uncertain. ** Divide every term by x under the radical divide by the x 2 15

16 30. Which of the following functions has a horizontal asymptote at y = 2? ** Recall the "shortcut" 1. If the degree (exponent) of the numerator is less than the degree of the denominator, the HA is y = 0 2. If the degree of the numerator is equal to the degree of the denominator, the HA is y = a/b (leading coefficients) 3. If the degree of the numerator is greater than the degree of the denominator, there is no HA (and there is a slant asymptote) A Degree of top Degree of bottom B Degree of top Degree of bottom C Degree of top Degree of bottom D Degree of top Degree of bottom 31. Find the horizontal asymptotes (if any) 16

17 32. Find the limit: ** Divide by x Use a graphing utility to graph the function then find the limit as x approaches infinity. 34. Find the limit ** Remember that the graph of sinx always oscillates between 1 and 1. Knowing that, a compound inequality can be set up Technically, we can move the 1/3 to the front of the limit notation (since it is a constant) 17

18 18