Mathematics 133 – Intensive Calculus for Science 1 Exam 3 – Things To Know November 27, 2000

General Information

As announced in the course syllabus, the third and last full-period midterm exam of the semester will be given in class on Friday, December 1. The format will be similar to that of the first two midterm exams. The exam will cover the material since the second midterm (Quizzes 6,7,8), including the following material from Chapter 4 sections 4,5,6, Chapter 5 sections 1,3,5, and Chapter 3 sections 1,2,3 of the text:

1) The Chain Rule for derivatives:

\frac{d}{dx}f(g(x)) = f′(g(x))g′(x)

2) Derivatives of trigonometric and inverse functions:

\frac{d}{dx} sin(x) = cos(x)
\frac{d}{dx} cos(x) = − sin(x)
\frac{d}{dx} ln(x) = \frac{1}{x}
\frac{d}{dx} arcsin(x) = \frac{1}{\sqrt{1 − x^{2}}}
\frac{d}{dx} arctan(x) = \frac{1}{1 + x^{2}}

Be able to use these in combination with the sum, product, quotient, and chain rules.

3) Critical points of functions, the First and Second Derivative Tests

4) Applied max/min problems – I will ask one very close to the ones in the handout given in class on Monday, November 20.

5) Left- and Right-hand sums for functions, the definite integral, area and average value of a function, the equation |RHS − LHS| = |f(b) − f(a)|Δx if f is either increasing or decreasing on the whole interval [a, b].

We will review for the exam in class on Thursday, November 30. If there is interest, I will also try to set up an evening session with Jodi Marani Wednesday or Thursday for other questions.

Sample Exam

I. Find the derivatives of each of the following functions. Say which rules you are applying in each case:

A) f(x) = (1 − x^{2})^{5/2} B) g(x) = \frac{cos(x)}{1−sin(x)} C) h(x) = arctan(x ln(x))

II. The following graph shows y = f′(x) (the derivative of f).

Use the information there to find the critical points of f, and classify each as a local maximum, local minimum, or neither. Note: I may also ask a question of this kind where the given information is a formula for f, and you must find f′(x), determine the critical points, and the sign of f′(x) between each consecutive pair of critical points to apply the First Derivative test and/or compute f′′(x) to apply the Second Derivative Test.

III. Holy Cross is building a new running track next to Fitton Field. It is to be the perimeter of the region obtained by putting together two identical semicircles on opposite sides of a rectangle. Due to the ongoing parking crunch on campus, the area surrounded by the track will be used as a parking lot when track practice is not being held. If the track is to be 500 yards long, determine the dimensions of the track to maximize the area of the parking lot.

IV. The following graph shows y = f(x).

A) Use the information from the graph to compute the left- and right-hand sums for f(x)

on [0, 3] with Δx = 1/2. B) Use the information from the graph to compute the integral ∫_{0}^{3} f(x) dx. C) Your answer from B should be negative. What does this mean in terms of areas?

Also see Problem Sets 6,7,8, the last three quizzes, and the Review Problems from the ends of Chapters 3,4,5 in the text for other practice problems.