AP Precalculus Notes: Unit 1 Topic 1.2 Rates of Change

From the Course & Exam Description (2023)

• Learning objectives:

  • 1.2.A Compare the rates of change at two points using the average rates of change near the points.
  • 1.2.B Describe how two quantities vary together at different points and over different intervals of a function.

• Essential Knowledge:

  • 1.2.A.1 The average rate of change of a function over an interval of the function’s domain is the constant rate of change that yields the same change in the output values as the function yielded on that interval of the function’s domain. It is the ratio of the change in the output values to the change in input values over that interval.
  • 1.2.A.2 The rate of change of a function at a point quantifies the rate at which output values would change were the input values to change at that point. The rate of change at a point can be approximated by the average rates of change of the function over small intervals containing the point, if such values exist.
  • 1.2.A.3 The rates of change at two points can be compared using average rate of change approximations over sufficiently small intervals containing each point, if such values exist.
  • 1.2.B.1 Rates of change quantify how two quantities vary together.
  • 1.2.B.2 A positive rate of change indicates that as one quantity increases or decreases, the other quantity does the same.
  • 1.2.B.3 A negative rate of change indicates that as one quantity increases, the other decreases.

Mr. Boone’s Perspective

• When you hear "rate of change," think slope of a line. Slope intuition from prior courses applies.

• Constant function example: f(x) = c (any real number). The rate of change over any interval where the output does not change is zero.

  • Graph example: f(x) = 9 (constant). The rate of change on any interval is 0.

• Linear function example: f(x) = mx + b

  • The function is always changing in a constant way; the rate of change is the slope, m.
  • For any two consecutive integer inputs, the change is constant: ∆f = f(x+1) − f(x) = m.
  • Calculation:
  • egin{aligned} riangle f &= f(x+1) - f(x) \&= m(x+1) + b - (mx + b) \&= mx + m + b - mx - b \&= m \end{aligned}

• Quadratic function example: f(x) = −x^2 + 4x

  • Rate of change is not constant; it depends on x.
  • Near the vertex, the rate of change is small; far from the vertex, it’s larger.
  • To characterize rate of change for non-constant functions, use the average rate of change over an interval.
  • Definition: The average rate of change of a function f on the interval [a, b] is the constant rate of change required to get from (a, f(a)) to (b, f(b)); it is the slope of the secant line joining these two points.
  • Formula (where m̄ denotes the average rate of change):
  • \bar{m} = \frac{f(b) - f(a)}{b - a}

• Key ideas:

  • The average rate of change depends on the chosen interval; it can vary dramatically with different intervals.
  • If the rate of change near a point is positive, increasing x near that point increases f(x); if negative, f(x) decreases as x increases.
  • Conceptual bridge to instantaneous rate of change: by making the interval smaller, the average rate of change approaches the instantaneous rate of change at a point (the derivative intuition).

• Example with f(x) = −x^2 + 4x on [0, 4]:

  • Compute f at endpoints:
  • f(0) = -0^2 + 4\cdot 0 = 0
  • f(4) = -(4)^2 + 4\cdot 4 = -16 + 16 = 0
  • Average rate of change on [0, 4]:
  • \bar{m} = \frac{f(4) - f(0)}{4 - 0} = \frac{0 - 0}{4} = 0
  • Geometric interpretation: the secant line through (0,0) and (4,0) has slope 0; this matches the average rate of change.
  • Partitioning into subintervals (equal length): [0,2] and [2,4]
  • On [0,2]: \bar{m}_{0,2} = \frac{f(2) - f(0)}{2 - 0} = \frac{4 - 0}{2} = 2
  • On [2,4]: \bar{m}_{2,4} = \frac{f(4) - f(2)}{4 - 2} = \frac{0 - 4}{2} = -2
  • Averaging the two subinterval rates (since equal-length subintervals): \frac{2 + (-2)}{2} = 0 which matches the larger-interval result.
  • Observations:
  • Rate of change is small near the vertex and larger far from it.
  • Left side (x < 2) has positive rate; right side (x > 2) has negative rate.
  • Approximation of rate at a point using a small interval around the point:
  • Example for x1 = 2 with width 0.1 centered at 2:
    • \bar{m}1 = \frac{f(x1 + 0.05) - f(x1 - 0.05)}{(x1 + 0.05) - (x_1 - 0.05)} = \frac{f(2.05) - f(1.95)}{0.1} = 0
  • Example for x2 = 10 with width 0.1:
    • \bar{m}_1 = \frac{f(10.05) - f(9.95)}{0.1} = \frac{-60.8025 - (-59.2025)}{0.1} = -16
  • Significance: positive rate near a point means f increases with x; negative rate means f decreases with x.

From Sullivan & Sullivan

• In words:

  • The symbol (\Delta) is the Greek capital delta and is read "change in."

• Find the Average Rate of Change of a Function:

  • In Section 1.5, the slope of a line is interpreted as the average rate of change.
  • To find the average rate of change of a function between two points on its graph, calculate the slope of the line containing the two points.

• Definition (Average Rate of Change):

  • If (a) and (b) are in the domain of a function (y = f(x)), the average rate of change of (f) from (a) to (b) is defined as
  • \text{Average rate of change} = \frac{f(b) - f(a)}{b - a}
  • The symbol (\Delta y) represents the change in (y), and (\Delta x) represents the change in (x).
  • The average rate of change is the change in y divided by the change in x.

• The Secant Line:

  • The average rate of change has a geometric interpretation: the line through ((a, f(a))) and ((b, f(b))) is the secant line.
  • Slope of the secant line (called (m_{sec})):
  • m_{sec} = \frac{f(b) - f(a)}{b - a}
  • The slope of the secant line equals the average rate of change from (a) to (b).

• Example 7 (f(x) = 3x^2):

  • Compute average rate of change on various intervals:
  • (a) From 1 to 3:
    • (f(3) = 3(3)^2 = 27), (f(1) = 3(1)^2 = 3)
    • \text{Average rate of change} = \frac{27 - 3}{3 - 1} = \frac{24}{2} = 12
  • (b) From 1 to 5:
    • (f(5) = 3(5)^2 = 75), (f(1) = 3)
    • \frac{75 - 3}{5 - 1} = \frac{72}{4} = 18
  • (c) From 1 to 7:
    • (f(7) = 3(7)^2 = 147)
    • \frac{147 - 3}{7 - 1} = \frac{144}{6} = 24
  • Conclusion: The function is increasing on the interval, e.g., on ([0, \infty)).
  • The positive average rates on these intervals reflect increasing behavior on the given domain.

• The Secant Line (Theorem):

  • The average rate of change of a function from (a) to (b) equals the slope of the secant line through ((a, f(a))) and ((b, f(b))) on its graph.

• Example 8 (g(x) = 3x^2 − 2x + 3):

  • (a) Find the average rate of change from (-2) to (1):
  • Compute values: (g(1) = 3(1)^2 - 2(1) + 3 = 3 - 2 + 3 = 4), (g(-2) = 3(-2)^2 - 2(-2) + 3 = 3(4) + 4 + 3 = 19).
  • Average rate of change:
  • \text{Average rate of change} = \frac{g(1) - g(-2)}{1 - (-2)} = \frac{4 - 19}{3} = \frac{-15}{3} = -5
  • (b) Find an equation of the secant line containing ((-2, g(-2))) and ((1, g(1))):
  • Slope is (m_{sec} = -5).
  • Using point-slope form with point ((-2, 19)):
  • y - y1 = m{sec} (x - x1)\quad\text{with}\quad (x1, y_1) = (-2, 19)
  • y - 19 = -5\,(x - (-2))
  • Simplify:
  • y - 19 = -5x - 10 \Rightarrow y = -5x + 9
  • (c) Interpretation: Graph the function and the secant line on the same screen (e.g., on a TI-84 Plus CE).

• Now Work Problems:

  • These are practice problems illustrating the definitions and secant-line interpretation of the average rate of change.

The Secant Line (Formal)

• The slope of a secant line between (a) and (b) on the graph of (y = f(x)) is:
• m_{sec} = \frac{f(b) - f(a)}{b - a}
• Theorem: The average rate of change of a function from (a) to (b) equals the slope of the secant line through ((a, f(a))) and ((b, f(b))).

Connections and Takeaways

• Rates of change connect algebra with geometry: average rate of change is a slope, i.e., the slope of a secant line.
• The concept generalizes the idea of slope from lines to general functions: as you shrink the interval, the average rate of change approaches the instantaneous rate of change (derivative) at a point.
• Positive vs negative rates: indicates whether the function value increases or decreases with increasing input over the interval.
• The interval [a, b] is essential: changing the interval changes the rate of change (even for the same function).

Quick Reference Formulas

• Average rate of change on [a, b]:

• \bar{m} = \frac{f(b) - f(a)}{b - a}

• Slope of the secant line through (a, f(a)) and (b, f(b)):

• m_{sec} = \frac{f(b) - f(a)}{b - a} = \bar{m}

• For the linear function f(x) = mx + b, the rate of change over any interval is constant and equals the slope m (as shown by the identity \Delta f = f(x+1) - f(x) = m when comparing consecutive integers).

• For a constant function f(x) = c, the rate of change over any interval is zero.

• Example recap (selected values):

  • f(x) = -x^2 + 4x on [0, 4]:
  • f(0) = 0,\, f(4) = 0
  • \bar{m} = \frac{0 - 0}{4 - 0} = 0
  • g(x) = 3x^2 - 2x + 3 on [-2, 1]:
  • g(-2) = 19, \ g(1) = 4
  • m_{sec} = \frac{4 - 19}{1 - (-2)} = -5

• Conceptual takeaway: the average rate of change is a bridge between function values at two points and the idea of a constant rate of change over an interval, captured geometrically by the secant line.