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7.1: Antiderivatives
Objectives:•To find the antiderivative of a function using the rules of antidifferentiation•To find the indefinite integral•To apply the indefinite integral
DERIVATIVES
• Up until this point, we have done problems such as: f(x)= 2x +7, find f’(x)
• Now, we are doing to do problems such as: f’(x)=2, find f(x)
• We do this through a process called antidifferentiation
Warm Up:
1. Find a function that has the derivative f’(x)=3x2+2x
2. Find a function that has the derivative f’(x)=x4-4x3+2
DEFINITION: ANTIDERIVATIVE
If F’(x) = f(x) then F(x) is an antiderivative of f(x)
F’(x) = 2x, then F(x) = x2 is the antiderivative of 2x (it is a function whose derivative is 2x)
Find an antiderivative of 6x5
2 antiderivatives of a function can differ only by a constant:
f’(x) = 2x g’(x)=2xF(x) = x2 +3 G(x)=x2-1
F(x)-G(x)= C
The constant, C, is called an integration constant
INDEFINITE INTEGRAL!!!!!
integral sign
f(x) integrand
dx change in x (remember differentials?!?!)
Be aware of variables of integration…
dxxf )(
If F’(x) = f(x), then = F(x) + C, for any real number C
F(x) is the antiderivative of f(x)
This is a big deal!!!!!!!
dxxf )(
Example:
Find the indefinite integral.
xdx2
Rules of Integration
Power Rule
Constant Multiple Rule(k has to be a real #, not a variable)
Sum or Difference Rule
1
1
n
xdxx
nn
dxxfkdxxfk )()(
dxxgdxxfdxxgxf )()()()(
Examples: Find the indefinite integral
1. 2. dxxx 232 xdxcos
3. 4. dtt 34 dxx
x
22
5. dxx22 5
More Rules…..
Cedxe xx
Ck
edxe
kxkx
Cxdxx
dxx ln11
Examples: Find the Indefinite Integral
dxx6
.1
dxxe x 23.2
dtett
2
12sec.3
Initial Value Problems
Find the function, f(x), that has the following:
1)2(;1
)('2
fxx
xf
4)2(;13)(' 2 fxxf
Find an equation of the curve whose tangent line has a slope of f’(x)=x2/3 given the point (1, 3/5) is on the curve.
Applications
1. An emu is traveling on a straight road. Its acceleration at time t is given by a(t)=6t+4 m/hr2. Suppose the emu starts at a velocity of -6 mph (crazy…its moving backwards) at a position of 9 miles. Find the position of the emu at any time, t.
(Acceleration due to gravity= -32 ft/sec2)
A stone is dropped from a 100 ft building. Find, as a function of time, its position and velocity. When does it hit the ground, and how fast is it going at that time?