Tangent Lines

Learn tangent slope, tangent-line equation.

Lesson 2 of 32 parts Practice Flashcards

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Tangent Slope

Objective: Explain tangent slope, apply the stated method, and check its conditions.

At a differentiable point, the derivative f′(a) is the tangent line’s slope. The tangent represents first-order local behavior rather than a line required to intersect the curve only once. Its equation uses both the slope and the actual point on the graph. A tangent can intersect a curve again away from the tangency point.

Formulas

Tangent slope = f′(a)

Worked examples

Try itFind the tangent slope to y=6x²+7 at x=7.Show answer

Step 1: The derivative is y′=2(6)x. Step 2: Evaluate at x=7. Step 3: The tangent slope is 84.

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Tangent Slope

Objective: Explain tangent slope, apply the stated method, and check its conditions.

Formulas

Tangent slope = f′(a)

Worked examples

Try itFind the tangent slope to y=6x²+7 at x=7.Show answer

Step 1: The derivative is y′=2(6)x. Step 2: Evaluate at x=7. Step 3: The tangent slope is 84.

Tangent-Line Equation

Objective: Explain tangent-line equation, apply the stated method, and check its conditions.

For differentiable f at a, the tangent line is y−f(a)=f′(a)(x−a). Evaluate the point and slope separately before expanding. An intercept is not generally f(a); it is the line’s output at x=0. Use the function value for the point and derivative for the slope.

Formulas

$y=f(a)+f′(a)(x−a)$

Worked examples

Try itThe tangent to y=x² at x=7 is y=mx+d. Find d.Show answer

Step 1: The point is (7,49) and m=2(7)=14. Step 2: Use y−49=14(x−7). Step 3: Expand: y=14x−49, so d=-49.

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