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?