AP Calculus Worksheet: Chapter 2 Review Part I

Transcription

1 AP Calculus Worksheet: Chapter 2 Review Part I 1. Given y = f(x), what is the average rate of change of f on the interval [a, b]? What is the graphical interpretation of your answer? 2. The derivative of a function f at x = a, written f '(a), is defined as What is the graphical interpretation of f '(a)? 3. State three interpretations of the derivative of a function at a point. 4. For any function f, we define the derivative function, f ', by 5. Given y = f(x), represent the derivative of f at x in three different ways. 6. Suppose f '(a) does not exist. State three reasons why this might occur.

2 2 WorksheetChapter2ReviewPartI.nb State the following derivative rules, both using symbols and words. 7. Derivative of a constant 8. Derivative of a linear function 9. Derivative of a constant multiple 10. Derivative of a sum or difference 11. The product rule 12. The quotient rule 13. The chain rule 14. If the graph of a function is concave upward on an interval what do you know must be true about the first and second derivatives on the same interval? What about if the graph is concave downward on an interval? 15. A function is continuous on an interval (a, b). For some number c, between a and b, the derivative is undefined and changes signs from positive to negative. Describe the point (c, f(c)).

3 WorksheetChapter2ReviewPartI.nb A function is continuous on an interval (a, b). Given a c b, f '(c) = 0 and f "(c) > 0. Describe the point (c, f(c)). Draw a portion of a graph on the interval (a, b) that meets each of the following conditions. 17. f'(x) is positive and f"(x) is positive. 18. f'(x) is negative and f"(x) = f'(x) is positive and f"(x) is negative. 20. f'(x) is negative and f"(x) is negative. 21. f'(x) is negative and f"(x) is positive. 22. f'(x) is positive and f"(x) is 0.

4 4 WorksheetChapter2ReviewPartI.nb 23. Sketch a graph that meets the following conditions. f '(c) is not defined. f '(x) > 0 for x < c and f '(x) < 0 for x > c. f "(c) is not defined. f '(x) > 0 for all x c. Complete the following. 24. If f ' > 0 on an interval, then f is 25. If f ' < 0 on an interval, then f is 26. If f " > 0 on an interval, then f ' is and the graph of f is 27. If f " < 0 on an interval, then f ' is and the graph of f is 28. Describe critical points of a function. 29. A function f with a continuous derivative has an inflection point at p if either of two conditions are met. State them. 30. What does the Extreme Value Theorem tells us about a continuous function on an interval [a, b]?

5 WorksheetChapter2ReviewPartI.nb What is rectilinear motion? 32. Suppose x(t) defines the position of an object on a line and the graph of v(t) cross the t-axis from above to below the axis. What just happened with the object? 33. Suppose an object moving on a straight line is defined by x(t). What can you say about the object if the velocity and acceleration of the particle at a point have the same sign? 34. Suppose you are looking at the graph of the velocity of a particle moving on a straight line. What point on the graph indicates where the particle has greatest velocity? What point on the graph indicates where the particle mas minimum velocity? What point on the graph indicates where the particle has the greatest speed? 35. For values of x near a, f(x) 36. Suppose you are approximating a function at a point using the tangent line to the graph near the point. If the tangent line approximation is slightly higher than the actual function value what does this tell you about the shape of the graph near that point? What is the tangent line approximation is slightly lower than the actual value? 37. Suppose f ' is continuous on (-, ). If f '(2) = 0 and f "(2) = -5, what can you say about f? If f '(6) = 0 and f ' (6) = 0, what can you say about f?

6 6 WorksheetChapter2ReviewPartI.nb 38. Given x 2 + y 2 = 25. What do we call this type of function? How is it different than other functions we have studied? Describe the graph of this function. Which x-values on the graph have only one tangent line to the graph? What is the slope of the tangent lines at these points? Every other x-value on the graph has two tangent lines associated with it. Why? What do you know about the slopes of the two tangent lines at each x-value?