8.1 supplemental HW 1.) For each of the following differential equations (i) draw its direction field; (ii) sketch the solution of the direction field that passes through the point (-2, 1); (iii) state the general solution to the differential equation. a.
Calculus Study Guide: Chapters 4-5
1. Exponential Functions a. b. x is any real # c. a = base 2. Exponential Growth a. b. Initial value c. 3. Derivative of the Exponential Function a. b. 4. Logarithmic Functions a. b. 5. Logarithmic Properties a. b. c. d.
Chapter 5 Review: Solutions
The Fundamental Theorem and the Antiderivative 1.
x d ln(t)dt = ln(x) A (x) = dx 1 d d 1 A (x) = A (x) = ln(x) = . dx dx x
2. 3. 4. 5.
True. True. False. False.
Areas of Regions 2. First, find the two intersection points. -x2 +
Chapter 5 Review: Solutions
U-Substitution 1. Let u = x2 + x so that du = (2x + 1)dx and solve for dx = (2x + 1)eu = eu du = e(x
2 +x)
du . (2x+1)
Substitute:
du (2x + 1)
|1 = e2 - e0 = e2 - 1 -1
2. Let u = tan(x) so that du = sec2 (x)dx and solve for dx
Circle T for True and F for False. 1.) lim! = () if () is defined.
PRACTICE PROBLEMS CHAPTER 2 DIFFERENTIATION
T 2.) lim! = means the limit exists and is a really big number. T 3.) () = slope of tangent line to y = f(x) at x = a. T 4.) If f is continuou
CHAPTER 8
(* means the problem is more likely to be in your final test and you'd better look at them first.)
PART ONE: TRUE/FALSE QUESTIONS
1. The decay of a radioactive substance is modeled by a differential equation.
T
F
2. Two fundamental questions con
Chapter 5 Review: Integration
U-Substitution
1 (2x -1
4
+ 1)e(x
2 +x)
dx
tan(x) dx cos2 (x)
0
2 3x2 +x dx 0 2x3 +x2 +4 e [ln(x)]2 dx x 1
The Fundamental Theorem and the Antiderivative 1. Let A(x) =
x 1
ln(t)dt. Find A (x).
x a
2. T/F: Let B(x) = 3. T/F:
Chapter 4 Review
Growth Problems 1. A population of bacteria is initially measured to be 400. The population grows at a rate of 10 percent per hour, that is, dP = 0.1P where P is population and t is time in hours. After how long dt is the population doubl
4.1 and 4.2 Review Material
Note 1. The material listed below is only meant as a guide. Every effort has been made to compile the most important aspects of Sections 4.1 & 4.2. However, the author does not claim the following list is entirely comprehensive
Calculus Study Guide: Chapters 4-5
1. Exponential Functions a. b. x is any real # c. a = base 2. Exponential Growth a. b. Initial value c. 3. Derivative of the Exponential Function a. b. 4. Logarithmic Functions a. b. 5. Logarithmic Properties a. b. c. d.
Calc Final Study Guide
Chapter 2
a. a function f has a Limit L as x approaches a from either side 2. A function f is continuous at x = a if: a. f(a) exists b. Limit as x goes to a of f(x) exists c. The limit is the same as the output 3. Difference quotien