Review
Sections 5.1-5.4 and 6.1-6.4
MA 1160/1161

1. Given a graph of velocity vs. time, how can you estimate the total distance traveled?

2. Given a table of values of velocity as a function of time, how can you estimate the total distance traveled?

3. How do you use rectangles to get an overestimate and/or underestimate of the area under a graph?

4. Using rectangles of a fixed width to estimate the area under a monotone function f(x) between a and b, what is the formula for the difference between the overestimate and the underestimate? Explain your formula graphically. (Why must the function be monotone?)

5. What is the graphical interpretation of the integral of a positive function? What if the function is negative over some intervals?

6. What is a Riemann sum? What is a ``left-hand sum''?

7. Under what circumstances can you conclude that a left-hand sum is an overestimate of the integral?

8. How do you compute the ``average value'' of a function over an interval?

9. If you integrate the rate of change of f, what do you get?

10. Given the rate of change of f(x), how could you compute the total change in f between x=a and x=b?

11. Given a formula for f'(x) how can you compute the total change in f(x) between two x values (say, x=a and x=b)?

12. Given a graph of f'(x) and two x-values a and b, how can you estimate f(b)-f(a)?

13. Given a graph of f'(x), how can you sketch a possible graph of f(x)? (Why is there more than one possible graph?)

14. What are the units of the integral, $$\int_{a}^{b} g(x) \; dx$$, in terms of the units of g and/or x?

15. What is the Fundamental Theorem of Calculus?

16. How can you use integrals to find the area between two curves?

17. How can you tell, from a graph of y=f(x), whether a definite integral $$\int_{a}^{b} f(x) \; dx$$ will be positive or negative?

18. What happens to the value of a definite integral when you interchange the limits of integration?

19. If two functions satisfy $f(x) \leq g(x)$ for all x in some interval, what can you say about their integrals? How can you explain this graphically?

20. Given a graph of f'(x) and a value for f(0), how can you estimate f(2)?

21. How can you ``split'' one integral into two?

22. Suppose you have two different antiderivatives of f(x). How are they related?

23. If you are given f'(t) and want to know about f(t), what does the sign ($\pm$) of f' tell you?

24. What is ``the indefinite integral''? How is it different from an antiderivative?

25. How is an indefinite integral different from a definite integral? How are they related?

26. For each of the differentiation formulas, what is the corresponding integration formula? (Exclude the product, quotient, and chain rules.)

27. If you're given a differential equation, what is meant by the ``general solution''? Why does it involve an arbitrary constant?

28. Given a differential equation and an initial condition (or a point on the graph), how do you find the solution? How do you check your answer?

29. Given position as a function of time, how do you find the velocity function?

30. Given velocity as a function of time, how do you find the position function?

31. Suppose you know that acceleration is constant. How can you find the velocity as a function of time? What additional information do you need (besides the constant rate of acceleration)?

32. Suppose you know that acceleration is constant. How can you find the position as a function of time? What additional information do you need (besides the constant rate of acceleration)?

33. How can you define a function in terms of an integral? What is the derivative of such a function?

34. How do you find the derivative of an integral with respect to its upper limit?

35. What is the second version of the Fun. Theorem of Calculus, and how can you use it to construct antiderivatives?

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