CALCULUS CHAPTER 6 NOTES SECTION 6-1 (Day 1...
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CALCULUS CHAPTER 6 NOTES SECTION 6-1 (Day 1) Indefinite Integrals Indefinite Integrals: If F is the antiderivative of f: β« () = + - C is called Some key antiderivatives: β« = β« = β« = β« = β« = β« = β« = β« = β« = β« = β« = β« = Key Reminder: REMEMBER TO BRING ALL CONSTANTS AND NEGATIVES
Transcript of CALCULUS CHAPTER 6 NOTES SECTION 6-1 (Day 1...
CALCULUS CHAPTER 6 NOTES
SECTION 6-1 (Day 1) Indefinite Integrals
Indefinite Integrals: If F is the antiderivative of f:
β« π(π)π π = +
- C is called
Some key antiderivatives:
β« ππ = β«π π
π=
β« ππ = β« πππ =
β« πππ π π π = β« πππ ππ π π =
β« πππ π π π = β« πππ ππ π π =
β« ππππ π π π = β« ππππ π π π =
β« πππ π πππ π π π = β« πππ π πππ π π π =
Key Reminder:
REMEMBER TO BRING ALL CONSTANTS AND NEGATIVES
Examples:
β«(π πππ π β πππ ππ) π π =
β«(π
π β π+ πππ ππ β πβππ) π π =
β« πππ π π π π =
ASSIGNMENT: Page 312 #3 β 6, 9, 10, 13, 15, 17, 19, 22,
CALCULUS CHAPTER 6 NOTES
SECTION 6-1 (Day 2) Solving Differential Equations
Recall what the differential form is of an equation:
π π
π π= πππ β πππ ππ +
π
π
Initial Conditions β when a point is given that lies
Example: Solve this differential equation:
π π = (πβπ
πβ ) π π
Given the initial condition: y(-1) = -5, find the original equation.
Example: Given π = πππ π½, find s(t) when v(0) = 0 and s(0) = -3.
ASSIGNMENT: Page 313 #25 - 27, 29, 31 β 34, 36, 41, 42
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CALCULUS CHAPTER 6 NOTES
SECTION 6-1 (Day 3) Slope Fields
SLOPE FIELDS:
Definition: A Slope Field is plot of short line segment with slopes f(x, y) such that:
π π
π π= π(π, π)
This is what a slope field looks like.
Sketch the possible solution to slope field given f(0) = 2.
On the axis below, sketch the slope field of the following differential equation:
π π
π π=
π
π
Now, solve the possible differential equation by separation of variables.
ASSIGNMENT: SLOPE FIELDS HANDOUT #1 β 16
CALCULUS CHAPTER 6 NOTES
SECTION 6-2 (Day 1) Substitution Method
Substitution Method -
Examples:
β«(ππ + π)π π π =
β« πβππ β π π π =
β«π π
(π β π)π=
β«ππ π
π π π =
REMINDERS:
1. All Constants
2. No Variables brought out
3. Never bring any variables (xβs) over to the du
ASSIGNMENT: Page 321 β 322 #2 β 4, 6, 8, 9, 13, 17, 24
CALCULUS CHAPTER 6 NOTES
SECTION 6-2 (Day 2) Substitution w/ Trig Functions
Examples:
β« πππππ π πππ π π π =
β« ππππ
π πβ
π
π πππππ π π =
Making a U-Substitution:
Example:
β« πππβπ
π πβ
π
ππ½ πππ ππ½ π π½ =
ASSIGNMENT: Page 321 β 322 #11, 14, 16, 18, 19, 21, 22, 34, 36, 37
CALCULUS CHAPTER 6 NOTES
SECTION 6-2 (Day 3) Separating Variables
Recall Solving a Differential Equation:
π π
π π= (π + π)(π + π)
1. Separate
2. Integrate
3. Add
4. Solve
5. Find C (If possible)
Solve the differential equation below by separating variables and find C given the initial value given by y(0) = 1.
π π
π π= (πππ π)ππ+πππ π
ASSIGNMENT: Page 322 #42, 43, 44
CALCULUS CHAPTER 6 NOTES
SECTION 6-3 Integration by Parts
Integration by Parts is derived by integrating the Product Rule.
When to Use:
Evaluate:
β« π πππ π π π
Choose: Derivative Antiderivative
Multiple Integration by Parts: (Called
β« πππβπ π π =
Choose: Derivative Antiderivative
ASSIGNMENT: Page 328 # 2, 15, 16, 19
CALCULUS CHAPTER 6 NOTE
SECTION 6-4 Exponential Growth and Decay
Recall the equation used to calculate an amount compounded continuously:
Substituting: y for A; and y0 for P:
The derivative of this equation with respect to t is:
π π
π π= π β π
So, anytime you see this equation, its antiderivative is:
Also, recall calculating the amount compounded using a fixed rate:
π¨ = ( +
)
Example: Suppose you deposit $1200 in an account that pays 4% annual interest. How much will you have 6 years later if the interest is:
a.) Compounded Continuously:
b.) Compounded Quarterly:
Radioactive Decay (Half-Life)
The half-life of a certain element is 25 days. If 100 grams of the substance is present
initially, use π = ππ πππ (where t is measured in days) law of exponential change formula to find the following:
a. Find the exact value of k.
b. How much of the substance remains after 42 days.
c. When will there only be 20 grams remaining?
ASSIGNMENT: Page 338 #1-4, 9, 12, 13, 25
CALCULUS CHAPTER 6 ASSIGNMENT SHEET
SECTION 6-1 (Day 1) Indefinite Integrals
ASSIGNMENT: Page 312 #3 β 6, 9, 10, 13, 15, 17, 19, 22
SECTION 6-1 (Day 2) Solving Differential Equations
ASSIGNMENT: Page 313 #27, 29, 31 β 34, 36, 41, 42
SECTION 6-1 (Day 3) Slope Fields
ASSIGNMENT: Slope Fields Handout #1-16
SECTION 6-2 (Day 1) Substitution Method
ASSIGNMENT: Page 321 β 322 #2 β 4, 6, 8, 9, 13, 17, 24
SECTION 6-2 (Day 2) Substitution w/ Trig Functions
ASSIGNMENT: Page 321 β 322 #11, 14, 16, 18, 19, 21, 22, 34, 36, 37
SECTION 6-2 (Day 3) Separating Variables
ASSIGNMENT: Page 322 #42, 43, 44
SECTION 6-3 Integration by Parts
ASSIGNMENT: Page 328 # 2, 15, 16, 19
SECTION 6-4 Exponential Growth and Decay
ASSIGNMENT: Page 338 #1-4, 9, 12, 13, 25
CHAPTER SIX REVIEW SHEET
CHAPTER SIX REVIEW SHEET
CHAPTER SIX TEST
