Understanding Calculus: Study of Small Changes in Mathematics

1. Definition of Calculus Calculus is the study of infinitesimally small changes. Take for example the path of the helicopter below. In describing its motion, we can take several points on its path and give its GPS location at that point and the time at which each measurement was taken. If only a few points are used and we never saw the airplane fly, we might decide that the airplane flew like this. 32.2837, -106.7482 10:42.11 am 32.4895, -106.2165 10:44.12 am 32.4895, -106.2165 33.2985, -105.2826 32.2837, -106.7482 33.2985, -105.2826 10:43.46 am

2. Definition of Calculus Calculus is the study of infinitesimally small changes. Take for example the path of the helicopter below. In describing its motion, we can take several points on its path and give its GPS location at that point and the time at which each measurement was taken. We might try using more points like this. But when we draw the ACTUAL path, we see that we miss what might be a very important change. We would need an infinite number of points to perfectly describe the motion of the airplane!

3. But how do we perform the math? Typically, we describe the location of the airplane as an equation or set of equations. The path below might have an equation something like this…

4. But how do we perform the math? So, how do we solve this for the velocity? We could try what some of you have learned in AP physics… and But what do we use for the change in x, yort? No matter which (or even how many) points we choose, we still do not describe the velocity completely!

5. Definition of Calculus We nee to use an infinite number of points and do a calculation for EVERY ONE OF THEM!!!!!!! Thankfully, Sir Isaac Newton and Gottfried Leibniz each developed a simple method to perform a single calculation on an equation that gives the same result as an infinite number of calculations for an infinite number of points. CALCULUS! There are two parts to calculus… First, a treatment of ratios of two changes (the change in one quantity over the change in another. (derivatives) Second, an infinite sum of all values of a quantity (integral).

6. Definition of a Derivative We call the limit of the ratio of a change in two quantities as the denominator goes to zero a derivative. Derivatives are the slope of a curve. For example, the instantaneous velocity is the derivative of the position with respect to time.

7. Slopes of curves are the derivative of the function that makes the curve. • The derivative of the line is shown below for various points.

8. The derivative of a sum is the sum of the derivatives. If two derivatives are multiplied, it is possible to cancel

9. There are two groups of derivatives that we will most often use in this class. Polynomials Trigonometric Functions

10. Definition of Integration Integration is the inverse of derivation. To find an integral we find the thing that, when a derivative is taken, we get the thing in the integral. The integral of a derivative is the thing in the integral. There are two types of integrals, definite and indefinite. Integrals may be thought of as infinite sums and they tell us the area under a curve.

11. Area under a curve is the integral. • The integral of the line between various points is shown in the figure below. positive area negative area

12. There are two groups of integrals that we will most often use in this class. (Other integrals are found in appendix D.) Polynomials Trigonometric Functions

13. The integral of a sum is the sum of the integrals. The process of substitution Definite integrals