1. Gottfried Wilhelm von Leibniz 1646 – 1716 Gottfried Wilhelm von Leibniz 1646 – 1716 Gottfried Leibniz was a German mathematician who developed the present day notation for the differential and integral calculus though he never thought of the derivative as a limit. His philosophy is also important and he invented an early calculating machine.

4. Example: Find the function whose derivative is and whose graph passes through. Notice that we had to have initial values to determine the value of C.

5. The process of finding the original function from the derivative is so important that it has a name: You will hear much more about antiderivatives in the future. This section is just an introduction. The process of finding an antiderivative is antidifferentiation.

7. Integration Symbol Integrand Variable of integration (dummy variable) Antiderivative Constant of Integration

9. Since acceleration is the derivative of velocity, velocity must be the antiderivative of acceleration. Example: Find the velocity and position equations for a downward acceleration of a m/sec 2 and an initial velocity of v 0 m/sec.

10. Since acceleration is the derivative of velocity, velocity must be the antiderivative of acceleration. Example: Find the velocity and position equations for a downward acceleration of a m/sec 2 and an initial velocity of v 0 m/sec. 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.

11. Since acceleration is the derivative of velocity, velocity must be the antiderivative of acceleration. Example: Find the velocity and position equations for a downward acceleration of a m/sec 2 and an initial velocity of v 0 m/sec. The initial position is s(0) = s 0.

13. Examples: 1)