Chapter 2, sections 2



On the final exam, I will focus on Calculus topics, though you will still need to be able to do algebra in order to solve for critical points, simplify, etc.

Since I will write your exam, studying your tests 1 – 3 is wise!

Chapter 1

Sec. 1.1—The Cartesian Plane and the Distance Formula

(Algebra)

Sec. 1.2—Graphs of Equations

(Algebra)

Sec. 1.3—Lines in the Plane and Slope

(Algebra)

1.4—Functions

(Algebra)

1.5–Limits

• be able to evaluate limits by direct substitution (when it works), by simplifying first and then substituting, by making a table, or by looking at the graph as needed

• understand one-sided limits

• be familiar with the properties of limits (pg 51) as needed

1.6–Continuity

• be able to determine the intervals on which a function is continuous

• identify whether a discontinuity is removable or not

Chapter 2 (sections 2.1 – 2.2)

2.1–Definition of the Derivative

• be able to find the derivative using the limit definition (one question on the test will require use of the limit definition—shortcuts will earn 0 pts on this question)

• understand what derivatives tell us

• be able to write an equation for a tangent line

2.2—Rules for Differentiation

• be able to differentiate using shortcuts

• practice re-writing functions so that shortcuts can be applied

• be able to write an equation for a tangent line

Formulas you need to know (from test 1 material):

[pic]

Chapter 2 (sections 2.3 – 2.7)

2.3 –Rates of Change: velocity and marginals

• know how to find instantaneous rate of change and average rate of change

• understand the relationship between position, velocity and acceleration

• be able to find units of a derivative function

2.4 –Product and Quotient Rules

• know them

• be able to use them and recognize when to use them

2.5 –Chain Rule

• know it

• be able to use it and recognize when to use it

2.6 –Higher Order Derivatives

• be able to calculate them (& understand meaning)

• be comfortable with notation

2.7 –Implicit Differentiation

• be able to find [pic] implicitly

• don’t forget to use product rule or quotient rule when necessary

Chapter 3 (sections 3.1 – 3.4)

3.1 –Increasing/Decreasing Functions

• be able to find where a function is increasing or decreasing (understand what the first derivative tells us about the original function)

• know what critical numbers are and how to find them

• Critical numbers occur when__________________________________________

3.2 –Extrema and the 1st Derivative Test

• know what extrema are (both relative and absolute)

• be able to find them using the first derivative test

3.3 –Concavity and the 2nd Derivative Test

• understand what the second derivative tells us about the original function

• know what “concavity” is and what “inflection points” are & how to find them

• be able to use the second derivative test to find extrema

3.4 –Optimization

• be able to optimize any quantity using either 1st or 2nd derivative test

• you should be able to write your own function if necessary (see homework & suggested problems for examples)

Formulas from test 2 (not given)

[pic]

[pic]

[pic]

you should also know things like the Pythagorean theorem, formulas involving rectangles and/or rectangular boxes, etc. I’ll provide formulas for circles, spheres, cylinders, cones, etc. if needed.

Chapter 3 (sections 3.6 – 3.8)

3.6 – Asymptotes

Relationship between asymptotes (both horizontal and vertical) and limits

3.7 – Curve Sketching: A Summary

• This section pulls together continuity, differentiability, extrema, concavity, inflection points, and asymptotes (see page 231 for sections referenced)

3.8 – Differentials and Marginal Analysis

• Compute differentials and use them to approximate error.

• Formula: [pic] (recall dx = (x)

Chapter 4 (sections 4.1—4.5)

4.1—Exponential Functions

(Algebra)

4.2—Natural Exponential Functions

(Algebra)

4.3—Derivatives of Exponential Functions

• Be able to find derivatives of exponential functions.

• Formulas to know:

[pic]

4.4—Logarithmic Functions

(Algebra review) – properties of logs can make differentiation easier!

4.5—Derivatives of Logarithmic Functions

• Be able to find derivatives of logarithmic functions.

• Formulas to know:

[pic]

4.6—Exponential Growth and Decay

• Review problems from this section. It’s mostly algebra review, but there are some questions involving calculus concepts.

Chapter 5 (sections 5.1—5.5)

5.1—Antiderivatives and indefinite integrals

• Know what “antiderivatives” and “indefinite integrals” are.

• Know the notation: [pic]

• Basic rules you need to know: (next page)

[pic]

**This last one is the “simple power rule”. Notice that it does not work for n = -1 (section 5.3 tells us how to deal with that).

5.2—The general power rule

• Know and be able to use the general power rule:

[pic]

(Again, this does not work for n = -1 —see section 5.3).

5.3—Exponential and Logarithmic integrals

• Know and be able to use the rules for exponential integrals:

[pic]

• Know and be able to use the rules for logarithmic integrals:

[pic]

5.4—Area and the fundamental theorem of calculus

• Be able to find area under a given graph. Recall that the area under f(x) between

x = a and x = b is given by [pic]

• Know that [pic], and be able to use this to find definite integrals.

5.5—The area of a region bounded by two graphs

• Be able to find the area between two graphs. You may need to find points of intersection first.

Chapter 6 (sections 6.1 and 6.2)

6.1—Integration by substitution

• Be able to integrate by substitution.

• Be able to solve definite integrals by substitution.

6.2—Integration by parts

• Be able to integrate by parts.

• I will give you the formula: [pic] on the cover page of your exam.

• Remember, if you try integration by parts and it makes your problem worse, try a different choice for u and dv.

Chapter 7 (7.1, 7.3 – 7.5?) – coverage will depend on what we are able to finish in class. We’ll discuss this further on the last day of class (review day).

7.1—The Three-Dimensional Coordinate System

• Finding distance between points in 3-dimensional space

• Finding midpoint between points in 3-dimensional space

• Finding equations of spheres

7.3 – Functions of Several Variables

• Evaluating functions of several variables

• Reading contour maps and associating them with 3D functions

7.4 – Partial Derivatives

• Finding partial derivatives; notation: [pic], etc. (see text pg 484)

• Evaluating partial derivatives (i.e. plugging in a point)

• Finding second partial derivatives

7.5 – Extrema of Functions of Two Variables

• Critical points of functions of two variables

• Second-partials test for relative extrema (pg 498)

• (Note: the First-partials test on pg 495 requires that you visualize the graph of the function in 3 dimensions; the Second-partials test does not require this).

Chapter 6 Review questions: pg 450-451 #1 – 34

Chapter 7 Review questions: pg 544-546 #1 – 10, 13 – 14, 27 – 28, 39 – 54, 63 – 70. [pic]

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