Math 124 Final Exam Review

 

 

This review contains a list of the topics we covered with suggested problems from the book. The odd numbered problems have answers at the back. Once you go through the list, make a note of what you think you should review more. Look at the final exam archive for more questions.

 

Parametric Curves

You should understand circular motion and be able to write down its parametric equation. You should also be able to compute dy/dx and d2x/dy2 for parametric curves.

Suggested Problems: Question 33 in Section 10.1 and Questions 11-16 in Section 10.2.

 

Limits and L’Hospital’s Rule

We started the quarter with limits and (almost) ended with L’Hospital’s Rule on limits. Some limits are straight forward to compute in the sense that you only need to plug in the number. Sometimes a limit does not exist. In many limit questions, we are not able to tell the answer in the first step. These limits are one of the Indeterminate Forms. The most common are  and  . To deal with an indeterminate quotient, we either do algebra tricks or use L’Hospital’s Rule. Look up the other indeterminate forms and how we deal with them.

Suggested Problems: Questions 3-20 in Chapter 2 Review, Questions 7, 9, 11, 13, 17, 45, 47, 53, 55, 59 in Section 4.4.

 

Differentiation Methods

There is a list of all the methods you need in Midterm 2 Review with suggested problems. Almost all the questions on the final will require differentiation at some point. Make sure you are really solid on this.

Suggested Problems: Do as many problems as you need to from Chapter 3, focusing on particular sections where you may have trouble. Also, do Question 39 in Section 3.5.

 

Equation of the Tangent Line and Linear Approximation

You should be able to write down the equation of the tangent line to y=f(x) at any x=a, equation of the tangent line to any implicit equation at a point (a,b) and equation of the tangent line to x=f(t) and y=g(t) at t=a.

In the case of explicit functions and implicit functions, we use the equation of the tangent line for approximating. Approach each approximation question as an equation of the tangent line question. In the very last step, you change = to ≈ to get your approximation.

Suggested Problems: Questions 57, 59 and 61 in Chapter 3 Review, Questions 25-32 in Section 3.5, Exercises 3-10 in Section 10.2.

 

Related Rate Problems

These are story problems where two (or more) quantities are changing with time. We relate them by an equation and differentiate that equation with respect to time. Use the rates given and asked (with their units) to help you name the variables. If you are still not sure what is changing (which should be a variable in your equation) and what remains constant (which should be a number or a constant like a in your equation), draw two pictures at two different times. Also, be careful with the sign of the rates: Is the quantity increasing or decreasing?

Suggested Problems: Questions 11, 16 (0.6 m/s), 20 (about 1.008 m/s), 25 in Section 3.9 and Question 97 in Chapter 3 Review.

 

Graphing Functions

Review Sections 4.3 and 4.5. Be very careful with the derivatives and make sure your first and second derivatives information match. For example, if your get x=3 as a critical point and f’ tells you it should be a max, f’’(3) should not be positive. If the derivative information does not agree, go back and recalculate f’ and f’’. In your homework in webassign, you did not sketch any curves, but picked an answer from a list. So, before the final, practice your drawing skills.

Suggested Problems: Question 7, 13, 15, 29, 31 from Section 4.5.

 

Optimization Problems

These are story problems where you come up with a function and its domain from the story to minimize and maximize. Once you have your function, you find the critical point. Then depending on what to domain looks like you do one of the following: If the domain is of the from a≤x≤b, you compare the function value at a, b and the critical point. If the domain is not of that form, you do a first or second derivative test to verify it is the min or the max the question is asking for.

Suggested Problems: Questions 17, 29, 31, 69 in Section 4.5 and Question 15 in Chapter 4 Review.

 

Theory

Most of the above topics and the questions from those topics use the derivative as a tool. You compute it and use it. But, do you understand what the derivative is? Answer the questions below like you are explaining to a friend.

1.   What is the (limit) definition of the derivative of f(x)? What does it mean for the derivative to exist or the function being differentiable? How do you use the definition of the derivative to compute the derivative of a function? Compute the derivative of f(x)=1/x using the definition of the derivative and compare your answer with what you would get by using the differentiation rule.

2.   If you are given a graph or y=f(x), can you compute the derivative at some x=a? How? Can you tell where the function is not differentiable from its graph? Can you graph its derivative y=f’(x)?

3.   How would you check if a multipart function is differentiable? How would you handle a function with absolute value?

4.   If you are given the graph of the derivative of a function, what can you say about the function? What can you say about the second derivative?

Suggested Problems: Questions 1, 43, 47, 48 in Chapter 2 Review, Questions 11, 19, 21 in Chapter 2 True-False Quiz, Questions 1, 3, 9 in Chapter 4 True-False Quiz and Question 18 in Chapter 4 Review.