Video Lesson The Tangent Line The Tangent Line
The tangent line to a curve at a specific point is a straight line that best approximates the slope of the curve at that exact point.Unlike a secant line, which passes through two points, the tangent line represents the instantaneous rate of change.
To find the slope m of the tangent line at a point (x₀, f(x₀)), we use the limit of the slopes of secant lines as x gets closer and closer to x₀:
m = lim (x → x₀) [(f(x) - f(x₀)) / (x - x₀)]
Explanation:
y − y₀: The vertical distance between a moving point and the fixed point.
x − x₀: The horizontal distance between them.
The Limit: As the distance x − x₀ shrinks to zero, the secant line becomes the tangent line.
Example Example: Slope of the Tangent Line
Find the slope of the tangent line to the curve y = x² at the point (2, 4).Step 1: Identify your values
x₀ = 2
y₀ = 4
y = x²
Step 2: Set up the formula
m = lim (x → x₀) [(f(x) - y₀) / (x - x₀)]
Step 3: Substitute the function for y
m = lim (x → 2) [(x² - 4) / (x - 2)]
Step 4: Factor the numerator
m = lim (x → 2) [((x - 2)(x + 2)) / (x - 2)]
Step 5: Cancel the common factor
m = lim (x → 2) (x + 2)
Step 6: Evaluate the limit
m = 2 + 2 = 4
Answer:
The slope of the tangent line at (2, 4) is 4.
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