MATH  RESOURCES Part 3

Calculus and Derivatives

Finding Derivatives:

Remember that Calculus is going to "account" for 20% of the test. So out of 25 questions, you can do the math and figure that around 5 questions are going to deal with finding limits and derivatives. Thus the following algebra I / II skills are necessary to complete this operation. The first section will deal with exactly how to find a derivative and how it is important. It may be benificient for you to review your math vocabulary and familiarize yourself with some of the terms. We'll take a run down of step by step analysis.

For those of you who aren't familiar with Calculus, a derivative is simple the slope of any give point along the curve of a line. Picture in your head the curve generated by x^2. It slopes upward both to the left and right. If you were to draw a line just tangent to it's vertex at the origin [0,0], what would the slope be? The answer is zero, naturally.

Example:   x^3 + 3x^2 - 7x Step 1: Multiply the exponent by the coefficient Step 2: Drop the x by one degree Step 3: Proceed to next term if needed Results in:   3x^2 + 6x - 7

If we wanted to find the slope of the tangent line at the x intercept of 1, all you do is plug in 1 into the derivative. 3(1)^2 + 6(1) - 7 = 3 + 6 - 7 = 2 The slope is positive 2

Let's say you wanted to find the slope at [1,1]. Unfortunately the answer isn't as easy, we're going to need the derivative.

Applications of Derivatives

Given the equation, find the following information

Equation: f(x) = 3x + 5

III. Find the critical points. (Set the derivative equal to zero). This finds where a curved line has a horizontal tangent, or where it stops turning and goes the other way

I. Find the Equation of the tangent line. (Find the derivative)

f(x) = 3x + 5 --->  f '(x) = 3

II. Find the slope of the tangent line at x = {-2, 0, 3}

f '(x) = 3. You can't set the derivative equal to zero. There is no turning point. Remember this is a straight line.

f '(x) = 3 is a constant! There is no value for 'x'. Think about it, 3x + 5 is a straight line, so the slope will never change!!

IV. Find the coordinates of the critical points. (Plug answer III into the original euqation).

Again, this problem does not exist. There is no critical point, thus there aren't any coordinates to go with it.

V. Determine if the point is a minimun or maximum

Again, this problem does not exist. There is no critical point, thus there aren't any coordinates to go with it.