The following is a list of questions on some topics from Calculus I. You should be familiar with all of these topics. If you are unable to answer the questions, or are unfamiliar with some topic, please see me right away.

1. What is the equation of a line? Give an example. What does the slope of a line measure geometrically? How do you compute the slope of a line? Give an example. What does the sign (+ or -) of the slope of a line tell you?

2. What is the equation of an exponential function? Give an example. What is the base of an exponential function? What do the graphs of the exponential functions look like? When is an exponential function increasing?, decreasing? Give examples.

3. What is the equation of a logarithm function? What is the base of a logarithm function? What do the graphs of the logarithm functions look like? When is a logarithm function increasing?, decreasing? What are the main algebraic properties of the logarithm functions? Give examples.

4. What does it mean for a function to be invertible? How do you test for invertibility of a function? What is the Horizontal Line Test? Give an example of an invertible function.

5. What are the sine, cosine, tangent functions? What are they measuring, geometrically, on a circle? What do the graphs of the sine and cosine functions look like? Are these functions invertible?

6. How are the inverse of the tangent and the sine functions defined? How do you restrict the domain? Sketch the graphs of the arctangent and the arcsine function.

7. What is the relationship between the graph of a function, f (x), and that of the functions f (x) + k, f (x + k), k*f (x), f (k*x), where k is a constant? Sketch examples demonstrating these relationships.

8. What is a secant line to a graph? What is the average rate of change of a function over an interval? What is the difference quotient of a function? Give several examples. Carefully and completely explain the relationship between these three concepts.

9. What is a tangent line to a graph at a point? What is the instantaneous rate of change of a function at a point? How do you use the difference quotient to compute the slope of the tangent line to a function at a point? Give several examples. Carefully and completely explain the relationship between these three concepts.

10. What is a limit of a function at a point, at infinity? How do you compute it? Give several examples. What are the algebraic properties of limits of functions? Give several examples. What is a one-sided limit? Give an example. What is the relationship between the value of a function at a point and the value of the limit of the function at that point?

11. What does it mean for a function to be continuous at a point -- give both a formal definition and explain the idea being captured by that definition. Give an example of a function which is not continuous at a point. What is the INTERMEDIATE VALUE THEOREM? What is the relationship between a function being continuous at a point and a function being differentiable at a point? Give an example, if possible, of a function which is one, but not the other; if it's not possible, explain why.

12. What are the derivatives of the basic functions -- polynomial, exponential, logarithmic, trigonometric, inverse trigonometric?

13. What are the algebraic properties of the derivative of functions --- the sum property, the constant multiple property, the product property, the quotient property? What is the chain rule? To what kind of function does the chain rule apply? Give many examples of derivatives of functions.

14. What is an implicit function? Give an example. How do you compute the derivative of a function defined implicitly? Give several examples.

15. What is logarithmic differentiation? When must it be used? Give an example of the use of logarithmic differentiation.

16. How do you derive a relationship between the rates of change of two functions, a so-called related rates problem? Give several examples of related rates problems and their solutions.

17. What are the relationships between the graph of a function and the graph of its derivative? When is the graph of a function increasing?, decreasing?, concave upward?, concave downward? What is a point of inflection? What is the relationship between each of these and the derivative of the function? Give several examples.

18. What is a critical point of a function? What is a local maximum/minimum of a function? Give several examples. What is the relationship between a critical point of a function and a local extremum?

19. What is an absolute maximum/minimum value of a function? What does the EXTERME VALUE THEOREM say? Give an example. How do you use it to find the absolute extremum of a function on a closed interval? Give an example.

20. What is the Second Derivative Test? Give several examples of its use. What strategies can you use when the Second Derivative Test is inconclusive?

21. Explain how to solve an applied optimization problem. Give several examples.

22. What is an indeterminate form? List several types of indeterminate forms. What is L'Hopital's Rule? For which types of indeterminate forms does L'Hopital's Rule apply? Give several examples of the use of L'Hopital's Rule. Who was L'Hopital and for what is he best known? How do you pronounce "L'Hopital"?

23. What is the MEAN VALUE THEOREM? Give both an algebraic and a geometric interpretation of this theorem.

24. What is an anti-derivative of a function? Give several examples. How many anti-derivatives does a function have? How do they differ? In terms of anti-derivatives, what is the relationship between the position, velocity, and acceleration of a particle? Give an example.

25. What is the subject matter of the integral calculus? How do you approximate the area under the graph of a function by means of right-hand and left-hand approximations? Give an example. Explain how these approximations can be used to find the exact value of the area under the graph of a function by means of a
limiting operation.

26. What is a Riemann sum? How do you use a Riemann sum to approximate the area under the graph of a continuous function on a closed interval? Give an example of a Riemann Sum. What is a Riemann Sum computing geometrically? How do you use a Riemann Sum to find the exact area under a graph by means of a limit operation?

27. What is a definite integral? What are the algebraic properties of a definite integral -- the sum property, the constant multiple property? Give an example of each type of property.

28. How do you compute a definite integral of a function by means of an anti-derivative? What is the EVALUATION THEOREM? When does the Evaluation Theorem apply? Give several examples.

29. What is the FUNDAMENTAL THEOREM OF CALCULUS? Give its precise statement. Give an example of its use.

30. What is an indefinite integral? Give an example. How do you compute an indefinite integral? Give several examples. What is the relationship between a definite and an indefinite integral?

31. What is the average value of a function on a close interval? What is the INTERMEDIATE VALUE THEOREM FOR DEFINITE INTEGRALS? Give an example. 

32. In a short essay, explain the relationships between the geometric, the algebraic, and the physical interpretation of the derivative and the definite integral. Be sure to treat both the discrete and the continuous case.