4.9 AntiderivativesWed Feb 4
Do Now
Find the derivative of each function
1)
2)
Antiderivatives• Antiderivative - the original function in a
derivative problem (backwards)
• F(x) is called an antiderivative of f(x) if F’(x) = f(x)
• Antiderivatives are also known as integrals
Integrals + C
• When differentiating, constants go away
• When integrating, we must take into consideration the constant that went away
Indefinite Integral
• Let F(x) be any antiderivative of f. The indefinite integral of f(x) (with respect to x) is defined by
where C is an arbitrary constant
Examples
• Examples 1.2 and 1.3
The Power Rule
• For any rational power
• 1) Exponent goes up by 1
• 2) Divide by new exponent
Examples
• Examples 1.4, 1.5, and 1.6
The integral of a Sum
• You can break up an integrals into the sum of its parts and bring out any constants
EX
Closure
• Hand in: Integrate the following function
• HW: p. 280 #1-2 11-23 odds
4-9 Integrals of Trig, e, lnxThurs Feb 5
• Do Now
• Integrate the following:
• 1)
• 2)
HW Review: p.280 #1, 2, 11-23 odds
• 1) 23)• 2)• 11)• 13)• 15)• 17)• 19)• 21)
Trigonometric Integrals• These are the trig integrals we will work
with:
Examples
• Ex 1.7
Exponential and Natural Log Integrals
• You need to know these 3:
Example
• Ex 1.8
You try
• Integrate the following:
• 1)
• 2)
• 3)
Closure
• Hand in: Integrate the following
• HW: p. 280 #3-9 odds 26-29 all 36
4-9 Integrals of the form f(ax)Fri Feb 6
• Do Now
• Evaluate the following integrals
HW Review p.280 3-9 26-29 36
• 3)• 5) 2sinx + 9cosx + C 36) 4lnx – e^x + C• 7)• 9) a-ii b-iii c-i d-iv• 26)• 27) 12sec x + C• 28)• 29) –csc t + C
Integrals of the form f(ax)
• We have now seen the basic integrals and rules we’ve been working with
• What if there’s more than just an x inside the function? Like sin 2x?
Integrals of Functions of the Form f(ax)
• If , then for any constant ,
• Step 1: Integrate using any rule
• Step 2: Divide by a
Examples
• Ex 1.9
You Try
• Evaluate the integrals
Closure
• Hand in: Integrate the following
• HW: p.281 #31-39 odds, 30 38
• Quiz Next Thurs
4-9 Finding original functions through integrating
Mon Feb 9• Do Now
• Integrate
• 1)
• 2)
HW Review p.281 #30-39
• 30)
• 31)
• 33)
• 35)
• 37)
• 38)
• 39)
Revisiting the + C
• Recall that every time we integrate a function, we need to include + C
• Why?
Solving for C
• We can solve for C if we are given an initial value.
• Step 1: Integrate with a + C
• Step 2: Substitute the initial x,y values
• Step 3: Solve for C
• Step 4: Substitute for C in answer
Examples
You tryFind the original function
Closure
• Hand in: Find the original function of
• HW: p.281-282 #47-61 odds
• 4.9 Quiz Thurs Feb 12
4-9 Working from the 2nd derivative
Tues Feb 10• Do Now
• Integrate and find C
• 1)
• 2)
HW Review p.281-2 #47-61• 47)• 49)• 51)• 53)• 55)• 57)• 59)• 61)
Finding f(x) from f’’(x)• When given a 2nd derivative, use both
initial values to find C each time you integrate
• EX: f’’(x) = x^3 – 2x, f’(1) = 0, f(0) = 0
Acceleration, Velocity, and Position
• Recall: How are acceleration, velocity and position related to each other?
Integrals and Acceleration
• We integrate the acceleration function once to get the velocity function– Twice to get the position function.
• Initial values are necessary in these types of problems
Example 1
• If a space shuttle’s downward acceleration is given by y’’(t) = -32 ft/s^2, find the position function y(t). Assume that the shuttle’s initial velocity is y’(0) = -100 ft/s, and that its initial position is y(0) = 100,000 ft.
Ex 2
• A car traveling with velocity 24m/s begins to slow down at time t = 0 with a constant deceleration of a = -6 m/s^2. When t = 0, the car has not moved. Find the velocity and position at time t.
Closure
• Hand in: Determine the position function if the acceleration function is a(t) = 12, the initial velocity is v(0) = 2, and the initial position is s(0) = 3
• HW: p.282 #63-69 odds• 4.9 Quiz Thurs Feb 12
4.9 ReviewWed Feb 11• Do Now
• If a ball is thrown up into the air and begins to fall, it has an acceleration function of a(t) = -32 ft/s^2. Find the position function if the initial velocity is v(0) = 0, and its initial position is s(0) = 20 ft
HW Review p.282 #63-69
• 63)
• 65)
• 67)
• 69)
Integral Quiz Review• What to know:
– Power Rule– Trig Rules (sinx, cosx, sec^2 x)– The two exponential rules– Ln x– Sums and differences of integrals– Integral of f(ax)– Solving for C
• 2nd deriv / Acceleration may be included in this section
Review
• Worksheet p.332 #1-24 27 29-32
#55-60 65-68
+C
Closure
• Journal Entry: What is integration? How are integrals and derivatives related?
• HW: Finish worksheet
• Quiz Thurs Feb 12
4.9 ReviewTues Feb 11
• Do Now
• Given f ’’(x) = -32, f ‘ (0) = 2, and f(1) = 5, find f(x)
HW Review: p.332 #5,7,10,11,12
• 5)
• 7)
• 10)
• 11)
• 12)
HW Review p.332 #15 16 19 21 23
• 15) -2cosx + sinx + C
• 16) 3sinx + cosx + C
• 19) 5tanx + C
• 21)
• 23) 3sinx - ln|x| + C
HW Review p.332 #27 29 31 34 39
• 27)
• 29)
• 31)
• 34) tan3x + C
• 39)
HW Review p.333 #55-60
• 55)
• 56)
• 57)
• 58)
• 59)
• 60)
HW Review p.333 #65-68
• 65)
•
• 66)
• 67)
• 68)
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