Rates of Change and Tangent Lines to Curves

Finding Slope and Equation of a Line

• Slope of a line containing two points:
  • Given by $m = \frac{\text{rise}}{\text{run}} = \frac{\Delta y}{\Delta x}$
  • Example: Find the equation of the line passing through (6, -1) and (3, 5).
    • Slope: $m = \frac{5 - (-1)}{3 - 6} = \frac{6}{-3} = -2$
• Equation of a line (point-slope form):
  • For a line with slope $m$ passing through point $(a, b)$, the equation is $y - b = m(x - a)$.
  • Using point (3, 5): $y - 5 = -2(x - 3)$

Average and Instantaneous Speeds

• Goals:
  • Find average and instantaneous speeds of an object.
  • Find the equation of a line tangent to a curve at a given point.
• Example: Driving on a highway, pass Mile Post 50 at noon and Mile Post 80 at 12:30.
  • Instantaneous speed at 12:10: Cannot determine without more information.
  • Average speed over the half-hour trip:
    • Average speed $= \frac{\text{distance traveled}}{\text{time elapsed}} = \frac{30 \text{ miles}}{0.5 \text{ hours}} = 60 \text{ mph}$
• Graphical representation:
  • Average speed is the slope of the secant line connecting two points on a position graph.

Calculating Instantaneous Speed

• Example: Position of a train is $s(t) = 3t^2$ for the first 20 seconds, where $t$ is in seconds and $s$ is in feet.
  • Find the instantaneous speed at $t = 4$ seconds.
  • Cannot use the average speed formula directly because $\frac{0}{0}$ is undefined.
• Method: Use time intervals that get closer and closer to zero.
  • Calculate average speed for each interval.
  • Instantaneous speed is the value the average speed approaches as the interval length approaches zero.
• Table for calculations:
  • Fix one end of the interval at 4 seconds.
  • $h$ = difference between the beginning point (4 seconds) and the ending point $(4 + h)$ seconds.
  • $\Delta t = h$
  • $\Delta s = s(4+h) - s(4)$
  • Average speed $= \frac{\Delta s}{\Delta t}$

• Guess: Instantaneous speed at $t = 4$ seconds is 24 feet per second.
• Exact Calculation:
  • Use a generic time interval of length $h$ from 4 seconds to $(4 + h)$ seconds.
  • Average speed $= \frac{s(4 + h) - s(4)}{(4 + h) - 4} = \frac{s(4 + h) - s(4)}{h}$
  • $s(t) = 3t^2$ so $s(4 + h) = 3(4 + h)^2$ and $s(4) = 3(4)^2 = 48$
  • $\frac{3(4 + h)^2 - 48}{h} = \frac{3(16 + 8h + h^2) - 48}{h} = \frac{48 + 24h + 3h^2 - 48}{h} = \frac{24h + 3h^2}{h}$
  • Factor out $h$: $\frac{h(24 + 3h)}{h}$
  • Cancel $h$: $24 + 3h$
  • Let $h$ go to zero: $24 + 3(0) = 24$ feet per second.
  • The instantaneous speed at $t = 4$ is 24 feet per second.

Graphical Interpretation

• Secant Lines:
  • Draw a line through the points on the curve.
  • The slope of the secant line gives the average speed over the time interval.
• Tangent Line:
  • As the time interval approaches zero, the secant line approaches the tangent line.
  • The tangent line touches the curve at a single point.
  • The slope of the tangent line equals the slope of the curve at that point.
  • The slope of the tangent line equals the instantaneous speed.

Summary

• Average speed approaches instantaneous speed as the length of the time interval approaches zero.
• Apply the same idea to the rate of change of any quantity with a formula.
• For any graph, the slope of the tangent line at a point equals the limit of the slopes of the secant lines fixed at that point.

General Example: Tangent Line to a Graph

• Find the equation of the tangent line to the graph of $f(x) = 1 - x^3$ at $x = 2$.
  • $f(2) = 1 - (2)^3 = 1 - 8 = -7$
  • Point-slope form: $y - (-7) = m(x - 2)$, where $m$ is the slope of the tangent line.
• Calculate the average rate of change over an interval of length $h$.
  • $\text{Slope of secant line} = \frac{f(2 + h) - f(2)}{h}$

$\frac{[1 - (2 + h)^3] - (-7)}{h} = \frac{1 - (8 + 12h + 6h^2 + h^3) + 7}{h} = \frac{-12h - 6h^2 - h^3}{h}$

• Factor and cancel $h$: $\frac{h(-12 - 6h - h^2)}{h} = -12 - 6h - h^2$
• To find the slope of the tangent line, let $h = 0$: $-12 - 6(0) - (0)^2 = -12$
• Equation of the tangent line: $y + 7 = -12(x - 2)$