4.9 AntiderivativesWed Jan 7
Do Now
If f ’(x) = x^2, find f(x)
Antiderivatives• Antiderivative - the original function in a
derivative problem (backwards)
• F(x) is called an antiderivative of f(x) if F’(x) = f(x)
• g(x) is an antiderivative of f(x) if g’(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
You try
Trigonometric Integrals• These are the trig integrals we will work
with:
Exponential and Natural Log Integrals
• You need to know these 2:
Example
• Ex 1.8
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
Note
• While this works for “basic” chain rule functions, it does not work for anything more than a linear ‘inside’
Examples
• Ex 1.9
Extra Do Now if needed – ignore this
• Do Now• Integrate• 1)
• 2)
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
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
Closure
• Find the original function f(x)
• HW: p.280 #1-71 odds
4-9 Anti-derivativesThurs Jan 8
• Do Now• Integrate and find C• 1)
• 2)
HW Review
Closure
• Journal Entry: What is integration? How are integrals and derivatives related? Why do we include +C?
• HW: Ch 4 Ap questions (AP4-1) All of them