Tangent Lines and Rates of Change for MATH 122
Exam Relevance for MATH 122
Core interpretation of derivative. Finding tangent line equations is commonly tested.
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Tangent Lines
A tangent line touches a curve at exactly one point and has the same direction as the curve at that point.
Think of it this way: Imagine you're driving on a curvy road. At any moment, you're heading in some direction. The tangent line is like an arrow showing which way you're pointed RIGHT NOW — not where you've been or where you're going, just this instant.
Drag the point along the curve and watch how the tangent line changes direction!
Rate of Change
Rate of change tells you how fast something is changing.
Quick Example: If you drive 100 km in 2 hours, your rate of change of position (speed) is:
$$\text{Speed} = \frac{100 \text{ km}}{2 \text{ hours}} = 50 \text{ km/h}$$
This is an average rate — your speed over the whole trip.
But what if you want your exact speed at one specific moment? That's where calculus comes in!
Average vs Instantaneous Rate of Change
Try it: Drag the green dot toward the blue dot. When they're separated, the line between them is a secant line showing the average rate of change. When the green dot reaches the blue dot, it becomes the tangent line — the instantaneous rate of change!
Average Rate of Change (Algebra)
The average rate of change of $f(x)$ from $x = a$ to $x = b$ is:
$$\text{Average Rate} = \frac{f(b) - f(a)}{b - a}$$
This is the slope of the secant line connecting two points on the curve.
Instantaneous Rate of Change (Calculus)
The instantaneous rate of change at $x = a$ is:
$$\text{Instantaneous Rate} = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$$
This is the slope of the tangent line at that point — also known as the derivative.
The Key Connection
Slope of tangent line = Instantaneous rate of change = Derivative
All three describe the same thing: how fast $f(x)$ is changing at exactly one point.
The Tangent Line Formula
The equation of the tangent line to $f(x)$ at the point $(a, f(a))$ is:
$$y = f(a) + f'(a)(x - a)$$
This is point-slope form with:
- Point: $(a, f(a))$
- Slope: $f'(a)$
Problem: Find the equation of the tangent line to $f(x) = x^2$ at $x = 3$.
Step 1: Find the point on the curve
$$f(3) = 3^2 = 9$$
The point is $(3, 9)$.
Step 2: Find the slope (derivative)
$$f'(x) = 2x$$ $$f'(3) = 2(3) = 6$$
Step 3: Write the tangent line equation
$$y = f(3) + f'(3)(x - 3)$$ $$y = 9 + 6(x - 3)$$ $$y = 9 + 6x - 18$$ $$\boxed{y = 6x - 9}$$
Problem: For $f(x) = x^2$, find:
- (a) The average rate of change from $x = 1$ to $x = 3$
- (b) The instantaneous rate of change at $x = 2$
Step 1: Find the average rate (Part a)
$$\frac{f(3) - f(1)}{3 - 1} = \frac{9 - 1}{2} = \frac{8}{2}$$ $$\boxed{4}$$
Step 2: Find the instantaneous rate (Part b)
$$f'(x) = 2x$$ $$f'(2) = 2(2)$$ $$\boxed{4}$$
Interesting: The average rate from 1 to 3 equals the instantaneous rate at 2 (the midpoint). For parabolas, this always works!
Problem: A ball is thrown upward. Its height after $t$ seconds is $h(t) = 20t - 5t^2$ meters. Find:
- (a) The average velocity from $t = 0$ to $t = 2$
- (b) The instantaneous velocity at $t = 1$
Step 1: Find the average velocity (Part a)
$$h(0) = 0, \quad h(2) = 20(2) - 5(4) = 20$$ $$\text{Average velocity} = \frac{20 - 0}{2 - 0}$$ $$\boxed{10 \text{ m/s}}$$
Step 2: Find the instantaneous velocity (Part b)
$$h'(t) = 20 - 10t$$ $$h'(1) = 20 - 10$$ $$\boxed{10 \text{ m/s}}$$
At $t = 1$ second, the ball is moving upward at exactly 10 m/s.
Average Rate of Change
The average rate of change of a function between two points. This equals the slope of the secant line connecting the two points on the curve.
Variables:
- $f(a)$:
- function value at the starting point
- $f(b)$:
- function value at the ending point
- $a$:
- starting x-value
- $b$:
- ending x-value
Instantaneous Rate of Change (Derivative Definition)
The instantaneous rate of change at a single point. This is the slope of the tangent line at that point, also known as the derivative.
Variables:
- $f(a)$:
- function value at the point
- $f(a + h)$:
- function value at a nearby point
- $a$:
- the x-value where we want the rate
- $h$:
- a small change in x (approaches 0)
Tangent Line Equation
The equation of the tangent line to a curve at a specific point. This is point-slope form where the slope is the derivative.
Variables:
- $y$:
- y-coordinate on the tangent line
- $f(a)$:
- function value at the point of tangency
- $f'(a)$:
- derivative (slope) at the point of tangency
- $a$:
- x-coordinate of the point of tangency
- $x$:
- any x-value along the tangent line
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