2.4 Rates of Change and Tangent Lines Devil’s Tower, Wyoming
Average Rate of Change Average Rate of Change = Amount of change divided by the time it takes. Or, where Δy = the amount of change and Δx = the time it takes. This idea is used to find the tangent of a curve at a certain point.
Remember, the slope of a line is given by: The slope at (1,1) can be approximated by the slope of the secant line through (4,16). We could get a better approximation if we move the point closer to (1,1) (i.e., (3,9) ). Even better would be the point (2,4).
The slope of a line is given by: If we got really close to (1,1), say (1.1,1.21), the approximation would get better still How far can we go? Slope of a Tangent
slope slope at The slope of the curve at the point is:
The slope of the curve at the point is: is called the difference quotient of f at a. If you are asked to find the slope using the definition or using the difference quotient, this is the technique you will use.
The slope of a curve at a point is the same as the slope of the tangent line at that point. In the previous example, the tangent line could be found using: If you want the normal line, use the negative reciprocal of the slope. (in this case, ). (The normal line is perpendicular.)
If it says “Find the limit” on a test, you must show your work! Example 1: Let a Find the slope at . Note: If it says “Find the limit” on a test, you must show your work!
Example 1 (cont.): Let b Where is the slope ?
Example 2: For y = x2 at x = -2, find the slope of the curve, the equation of the tangent, and an equation of the normal. Then draw the graph of the curve, tangent line, and normal line on the same graph.
Example 2 (cont.) f(x) = x2 at x = -2 So, now we know y = -4x + b, but we do not know the value of b.
Example 2 (cont.) To find the value of b, and hence the equation of the tangent, use y = mx + b to find the y-intercept. At the given value of x = -2, y = (-2)2 or 4. So, substituting into y = mx + b, gives 4 = -4(-2) + b, which means b = -4. Therefore, the tangent line has an equation of y = -4x – 4 by substituting the values for m and b into y = mx + b.
Example 2 (cont.) Since the normal line is perpendicular to the tangent line, the slope is the opposite reciprocal of -4 or ¼. Also, the normal line goes through the same point as the tangent line or (-2, 4). So, use y = mx + b to find the y-intercept of the normal line. y = ¼x + b 4 = ¼(-2) + b 4 = -½ + b b = 9/2 This give an equation for the normal line of y = ¼x + 9/2.
Example 3 Free Fall. A rock breaks loose from the top of a tall cliff and falls 16t2 feet after t seconds. How fast is it falling 3 seconds after it starts to fall?
Review: p velocity = slope These are often mixed up by Calculus students! average slope: slope at a point: average velocity: And, so are these! instantaneous velocity: If is the position function: velocity = slope p