Differential and Integral Calculus

2011 Summer Session Two

This is the course webpage for Spencer Unger's calculus class.

Syllabus

Homework 1 Due June 29th

Section 1.1: 3-6,16,24,26,46
Section 1.2: 38,44
Section 1.3: 4,8
Section 1.4: 2,4,24,30,46

Homework 2 Due July 6 

Section 1.4 30,46
Section 1.5 14,16,18,26,28,36,37,38,46
Section 1.6 14,16,18,20,21,28
Section 2.1 16,18
Section 2.2 4,12,20,21,22,23,41 
Section 2.3 6,8,16,18,22,28,32,36,38 (No need to graph 28)
Section 2.4 4,6,12,16,30,34,42
Section 2.5 4,8,16,22,28,30

Test 1 July 8th

The test is over the material up through July 5th. The Chain Rule is the last topic that will be covered. Here are the objectives for the test.

Be able to determine the domain and range of a function

Evaluate limits or show that a limit does not exist. A known limit like that of sin(x)/x as x approaches 0 may be used.

Be able to differentiate functions involving algebraic and trigonometric functions. Know the rules of differentiation, power rule, product rule, etc.

Determine a derivative using the definition of derivative.

Know the Intermediate Value Theorem and its applications. Be able to state your conclusion in a complete sentence

Determine the equation of the tangent line to the graph of a function at a point.

On the test you will be asked to do one or two things off of the following list.

  1. Explain the intuitive definition of the limit.
  2. Explain intuitively what it means for a function to be continuous.
  3. Show that a given polynomial is continuous on the whole real line using limit laws as justification.
  4. Explain what it means for the line y=L to be a horizontal tangent line to a function f.
  5. Derive the formula for the product rule.
  6. Derive the derivative of sin(x).
  7. Derive the derivative of tan(x).
  8. Derive the formula for the quotient rule from the product rule and the chain rule. (Please ignore this item. I did not get to it in class. Changed 10:30AM July 6th)

Review Problems

Assigned homework problems are always good review. 

Section 1.1 29-36 Though the instructions don't ask for it you should be able to find the range of these functions too.
Section 1.4 11, 18, 21, 23, 34, 43-46
Section 1.5 13-16,35-40,41(a),42(a)
Section 1.6 18-21,28-31
Section 2.2 17,20-23 
Section 2.4 38-42
Chapter 2 Review 13-22,30,31,33,35,37,43,44,47-49,51-56,59,60

Homework 3 Due July 13th

I suggest waiting until after Test 1 to start this.

Section 2.6 7,8,12,18,22
Section 2.7 6,10,14,18
Section 2.8 12,14
Section 3.1 15,19,24,28
Section 3.2 11,12,14,21,23,24,48,52,62,69,72
Section 3.3 3,4,14,20,30,32,46,48
Section 3.4 4, 12
Section 3.5 3,4,5,7

Homework 4 Due July 20th 

3.5 18,20,24,26
3.6 2,4,10,12,26,30
3.7 2,4,6,8,20,22,24,30,36
4.1 4,8,24,34,36,40,46
4.2 12,13,16,32,34
4.3 2,4,8,9,13,15,16,49
4.4 2,12,14,28,34,38,39

I may or may not add optimization problems to this assignment. We'll see how fast we go. 7/19 Optimization problems will appear on the next assignment.

Test 2: Friday 7/22

Objectives for the test

There will be no linear approximation or exponential growth/decay problems on the test.

  1. Implicit differentiation. Be able to take derivatives of graphs defined implicitly.
  2. Related Rates. Be able to do problems from section 2.7.
  3. Know the definition of e (the base of the natural log.)
  4. New functions. Know the properties of the following functions, exponentials, logs, inverse trig and hyperbolic trig. Properties you should know for each function: definition, domain, range, appropriate limits, general shape of graph, algebraic properties and derivative.
  5. Inverse functions. Know the definition of 'one-to-one' and its relation to the existence of an inverse function. Know how to find the inverse of a function algebraically if it is possible. Know that logs and exponential functions are inverses of each other.
  6. Know how to evaluate a limit using L'Hopital's rule. This includes knowing all of the indeterminate forms mentioned in class and knowing how to manipulate them in order to apply L'Hopital.
  7. Be able to find the maximum and minimum values of a function on a closed interval.
  8. Know the Mean Value Theorem and its applications.
  9. Be able to find the local maximums and minimums of a function defined on the real line (or an open interval). This includes the first and second derivative tests.
  10. Know how to sketch the graph of a function using the guidelines given in class
  11. Know the defintions of the following concepts:
    1. Local maximum and minimum
    2. Absolute maximum and minimum
    3. Critical point
    4. Concave up and Concave down
    5. Point of inflection
    6. Horizontal and Vertical Asymptote

Review Problems (Always start by going over what you missed on the homework.) 

Section 2.6 3,5,9,11,17,18
Section 2.7 3-8, 15-17
Section 3.1 15,16,23,27,30
Section 3.2 21,23
Section 3.3 6,7,9,11,17-20,37,38
Section 3.5 1-4, 17-20 
Section 3.6 26-29
Section 3.7 1,4,5,6,21-25
Section 4.1 33-36,41,45,47,48
Section 4.2 11,13,28,34,35
Section 4.3 1-8
Section 4.4 3,4,9,10,29,33,35,43

Homework 5 due Thursday July 28th 

Section 4.5 4,6,10,21,34
Section 4.7 1,7,9,19,20,41
Section 5.2 15,19,23,35
Section 5.3 6-8,21,22,27

Assignment 6 due August 3rd 

Section 5.4 5,8,14  (Use the hint for 13.)
Section 5.5 3,4,8,10,22,26,28,30,34,36,38
Section 6.1 3,4,9,12,14,17,18,21,25,33
Section 6.2 2,6,12,22,37-39
Section 7.1 7,10,14,32

I may add to this assignment. Stay tuned. Section 6.2 added 7/29 11AM.

Objectives for the final exam

  1. Optimization. Be able to solve optimization word problems as in Chapter 4 Section 5. More specifically, be able to translate a written problem in to relevant equations and know the relevant techniques of finding maximums and minimums.
  2. Know how to find a function given its derivative and an initial value. Know the applications of antiderivatives to physics from Chapter 4 Section 7.
  3. Know the definitions associated with constructing a Riemann Sum. Know how to set up and compute Riemann Sums using equal subintervals and right (or left) endpoints.
  4. Know the properties of the definite integral listed in Chapter 5 Section 2. Know the relationship between the definite integral and area under the curve.
  5. Know the Evaluation theorem and the Fundamental theorem of Calculus and their applications.
  6. Know the antiderivatives of all of the common functions discussed in class.
  7. Know the techniques of u-substitution, integration by parts and trigonometric substitution. Be able to recognize the appropriate technique for a given integral.
  8. Know how to find the area enclosed by a given set of curves.
  9. Know and understand the following theorems/definitions:
    1. Partition of a closed interval [a,b]
    2. Sample points from a given partition
    3. The Riemann Sum associated with a given function, partition and set of sample points
    4. The definition of the definite integral of a function on an interval
    5. Theorems 3 and 4 from chapter 5 Section 2
    6. The Fundamental Theorem of Calculus (and the Evaluation theorem)
  10. Know the derivations of the antiderivates of the following functions: ln(x),tan(x),sec(x). Know how to derive the reduction formula for sin^{n}(x).

Review Problems. Start by going over everything on the last two homeworks, then work on the following list. 

Section 4.5 2,3,8,9,24,27
Section 4.7 19,20,37,40,41
Section 5.2 16,20,21,26,27 (For #27 you don't need to use a computer algebra system, just do the setup.)
Section 5.3 7-9,17,19,35
Section 5.4 5,6,9,12,13
Section 5.5 11,14,18,20,32,35-37
Chapter 5 Review 31,32
Section 6.1 5,7,10,14,19,23,25
Section 6.2 5,13,19,25,27,41,43
Section 7.1 5,6,9,16