Anti-Derivatives4.2 Revisited
These two functions have the same slope at any value of x. Functions with the same derivative differ by a constant.
so: Instead of taking a function and finding it’s derivative, let’s take a known derivative and see if we can back track to the original function. Find the function whose derivative is and whose graph passes through. In other words, let’s do things backwards. We have f and we want to find f. Wait! Where did the C come from?
These two functions only differ by a constant, but they have the same derivative. So working backwards from 2x, we have to allow for a constant C because we can’t know the exact value of C until we know more about the original function.
so: could be by itself or could vary by some constant. Instead of taking a function and finding it’s derivative, let’s take a known derivative and see if we can back track to the original function. Find the function whose derivative is and whose graph passes through. In other words, let’s do things backwards. We have f and we want to find f.
Instead of taking a function and finding it’s derivative, let’s take a known derivative and see if we can back track to the original function. Notice that we had to have initial values to determine the value of C. Find the function whose derivative is and whose graph passes through. so: In other words, let’s do things backwards. We have f and we want to find f. Plug in ( 0, 2)
The process of finding the original function from the derivative is so important that it has a name: Antiderivative A function is an antiderivative of a function if for all x in the domain of f. The process of finding an antiderivative is antidifferentiation. You will hear much more about antiderivatives in the future. This section is just an introduction.
Many anti-derivatives like sin(x) and cos(x) are intuitive. For polynomials, we need a power rule as we did for derivatives: Power Rule for derivatives: Power Rule for anti-derivatives: 1.Multiply by the exponent 2.Subtract 1 from the exponent. 1.Add 1 to the exponent 2.Divide by the new exponent Eventually, you will have to use the notation for anti- derivatives which looks like this:
Example 7b: Find the velocity and position equations for a downward acceleration of 9.8 m/sec 2 and an initial velocity of 1 m/sec downward. Velocity is the anti-derivative of acceleration
Example 7b: Find the velocity and position equations for a downward acceleration of 9.8 m/sec 2 and an initial velocity of 1 m/sec downward. Since velocity is the derivative of position, position must be the antiderivative of velocity. The power rule in reverse: Increase the exponent by one and multiply by the reciprocal of the new exponent.
Example 7b: Find the velocity and position equations for a downward acceleration of 9.8 m/sec 2 and an initial velocity of 1 m/sec downward. The initial position is zero at time zero.
The process of finding the original function from the derivative is so important that it has a name: Antiderivative A function is an antiderivative of a function if for all x in the domain of f. The process of finding an antiderivative is antidifferentiation. Problems like these are called Initial Value Problems because an initial value is needed to find C.