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)