1.2: Rate of Change by Equation, Graph, or Table [AP Calculus AB] Objective: Given a function y = f(x) specified by a graph, a table of values, or an equation, describe whether the y-value is increasing or decreasing as x increases through a particular value, and estimate the instantaneous rate of change of y at that value of x. Function Terminology and Types of Functions Function Terminology of f(x) f is the name of a function. (Note: other letters can also be used to name functions. For example d can be used to name a function that models distance.) x is the independent variable or specific x-value that will be substituted into the function. f(x) is the dependent variable or specific y-value that corresponds to the substituted x-value. Types of Functions Type General Equation *Variables other than x are constants Particular Equation Linear f(x) = mx + b f(x) = 2x 1 Particular Graph Quadratic f(x) = ax 2 + bx + c f(x) = x 2 + 1 Polynomial (odd) f(x) = a n x n ax 2 + bx + c f(x) = x 3 6x 2 9x 6
Polynomial (even) f(x) = a n x n ax 2 + bx + c f(x) = x 4 Exponential f(x) = ab x f(x) = e x Rational f(x) contains (polynomial) (polynomial) f(x) = x x 1 Absolute Value f(x) contains polynomial f(x) = x + 1 1 Trigonometric or Circular f(x) contians sin x, cos x, tan x, csc x, sec x, or cot x f(x) = 2 sin x Analyzing a Function Using its Derivative The derivative of a function f(x) at x = a is the instantaneous rate of change of f(x) with respect to x at x = a. It is found: Numerically, by taking the limit of the average rate over the interval from a to c as a approaches c. (Note: a limit is an arbitrary number that f(x) approaches) f(x) f(a) x a f (a) = lim x a Graphically, by finding the slope of the line tangent to the graph at x = a
The derivative f (x) can help you describe characteristics of the original function f(x). If f (a) is positive, then f(a) is increasing at x = a. If f (a) is negative, then f(a) is decreasing at x = a. If f (a) is zero, then f(a) is neither increasing nor decreasing at x = a. Typically, if f (a) is greater than 1, then the rate is fast and f(a) is changing quickly at x = a. Typically, if f (a) is less than 1, then the rate is slow and f(a) is changing slowly at x = a. (Note: classifying the rate as fast or slow depends on the context of the equation.) Example 1: Using the Derivative to Analyze a Function (Increasing/Decreasing, Fast/Slow) The figure shows the graph of a function. At x = a, x = b, and x = c, state whether y is increasing, decreasing, or neither as x increases. Then state whether the rate of change is fast or slow. Example 2: Using a Function s Graph to Determine the Derivative The figure shows the graph of a function that could represent the height, h(t), in feet, of a soccer ball above the ground as a function of time, t, in seconds since it was kicked into the air. a. Estimate the instantaneous rate of change of h(t) at time t = 5.
b. Give the mathematical name of this instantaneous rate, and state why the rate is negative. Example 3: Using a Function s Equation to Determine the Derivative The figure shows a graph of P(x) = 40(0.6 x ), the probability that it rains a number of inches, x, at a particular place during a particular thunderstorm. a. The probability that it rains 1 inch is P(1) = 24%. By how much, and in which direction, does the probability change from x = 1 to x = 1.1? (What is the average rate of change from 1 inch to 1.1 inches? Make sure to include units in your answer.) Why is the rate negative? b. Write an equation for r(x), the average rate of change of P(x) from 1 to x. Make a table of values of r(x) for each 0.01 unit of x from 0.97 to 1.03. Explain why r(x) is undefined at x = 1.
c. The instantaneous rate at x = 1 is the limit that the average rate approaches as x approaches 1. Estimate the instantaneous rate using information from part b. d. Estimate the instantaneous rate at x = 1 by using a value of your choice for x (just as performed in Section 1-1). Example 4: Using a Function s Table of Values to Determine the Derivative A mass is bouncing up and down on a spring hanging from the ceiling. Its distance, y, in feet, from the ceiling is measured by a calculator distance probe each 1/10 s, giving this table of values, in which t is time in seconds. a. How fast is y changing at each time? i. t = 0.3 ii. t = 0.6 iii. t = 1.0 b. At time t = 0.3, is the mass going up or down? Justify your answer.