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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