Review
Sect's 1.11, 4.5-4.8, 5.1-5.3, 5.5, 5.6
MA 160

1. What does it mean for a function to be continuous on a given interval? (give at least 2 answers)
2. Name some functions which are continuous and some which are not continuous at some point.
3. What is a limit? How is it different from the function value at a point? How can you deal with limits of rational functions? Under what circumstances can you use l'Hospital's rule?
4. What are the basic differentiation rules (formulas)? How do these change if you replace x with a function of x?
5. How do you know when to use each of the rules (formulas) for differentiation? Compare: power rule vs. exponential functions, product rule vs. chain rule.
6. Try to write each of the differentiation formulas in terms of the function g(t) instead of f(x).
7. How can you check a derivative formula using your calculator?
8. What are the steps you use to find the formula for the line tangent to a curve at a point?
9. If the derivative of f(x) is positive over an interval, what does that tell you about f itself?
10. Suppose you want to divide up the x-axis into regions where f' > 0 and where f' < 0. How do you decide where the divisions go? How do you decide the sign of f' on each of the regions?
11. If f''(x) is positive over an interval, what does that tell you about f itself?
12. Suppose you want to divide up the x-axis into regions where f'' > 0 and where f'' < 0. How do you decide where the divisions go? How do you decide the sign of f'' on each of the regions?
13. Give an example of the graph of a function f(x) for each of the following scenarios:
    • f'(x)>0 and f''(x)>0
    • f'(x)>0 and f''(x)<0
    • f'(x)<0 and f''(x)>0
    • f'(x)<0 and f''(x)<0
14. How does implicit differentiation work? Practice by differentiating (your answer for dy/dx should be the formula for the derivative of .)
15. What is a critical point?
16. What is a local maximum? A local minimum? A global (absolute) maximum? A global (absolute) minimum? How do you go about finding these using derivatives? How can you check you answer?
17. What is an inflection point? How is it related to the rate of change of slope?
18. Explain three ways to check whether a critical point is a local maximum/minimum or neither (using the first derivative, the second derivative, or the function itself).
19. Sketch the graph of a function which has a critical point that is NOT a maximum or a minimum.
20. Sketch the graph of a function which has , but which does NOT have an inflection point at .
21. Suppose we have a point at which the SLOPE of f is a maximum. What can we say about ?
22. When can you have a critical point at which the first derivative is NOT zero?
23. If you have a function of one variable, where do you look for the global maximum and minimum? Why?
24. What are the three questions Tami asks you to answer when dealing with any optimization (max. or min.) problem?
25. What is your game plan for attacking a word problem?
26. Suppose you are given the graph of a function $f(x)$ and a point on the graph. If this point is the ``first guess'', $x_0$, for Newton's method, show graphically how the second guess is found.
27. Suppose you are trying to find the solution to an equation. How would you write the equation so that you can use Newton's method? Once you've written the equation in the form $F(x)=0$ and made a first guess, what is the formula for getting the next guess for the solution?
28. In which situations could Newton's method fail? Why?

Besides thinking about the questions above, here are some other ways to study for the test:

• Review the worksheets covering this material.
• Practice symbolic differentiation using the random quiz generator (with answers) on Tami's MA160 web page (http://www.math.mtu.edu/ trolson/calc1.html).
• Read through your old lab notebooks, concentrating on the ideas rather than the computations. Ask yourself: ``What did we do here? Why?''
• Review your quizzes on this material.
• Review your homework on this material.
• Read through the examples in the book, and after going through each try to summarize the procedure or idea in a few sentences.
• Memorize formulas that you will need.
• Read through the review problems in the book and decide how you would approach each one.