Tangent Lines and Slopes

Business Calculus · Axiom Academy

LESSON Tangent Lines and Their Equations Finding the line that best approximates a curve at a point The Derivative Gives the Slope Now that we know how to find derivatives, we can use them to find the equation of the tangent line to a curve at any point. If f is differentiable at x = a , then the tangent line to the graph of f at the point (a, f(a)) has slope equal to f'(a) . This is the geometric meaning of the derivative: it tells us the slope of the tangent line at any point on the curve. The tangent line "just touches" the curve at one point and has the same direction as the curve at that point. To write the equation of any line, we need two things: For a tangent line to y = f(x) at x = a : This is the point-slope form of the tangent line equation. We can also solve for y to get slope-intercept form: Example: Find the tangent line to f(x) = x^2 at x = 3 Find the point: Calculate f(a) f(3) = 3^2 = 9 , so the point is (3, 9) Find the derivative: f'(x) f'(x) = 2x Find the slope: Calculate f'(a) f'(3) = 2(3) = 6 Write the equation: Use point-slope form Move the slider to see how the tangent line changes as you move along the curve f(x) = x^2 . Business Application: Local Linear Approximation Tangent lines have a powerful practical use: they provide linear approximations near a point. Near x = a , we can approximate: The closer x is to a , the better the approximation! A company's total cost function is C(x) = 500 + 20x + 0.01x^2 dollars for producing x units.

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