Combine a Derivative Slope with a Point
A tangent line at an input passes through that function point and has derivative slope there.
About 25 minutes, plus any extended practice
Supplied materials
- Paper or a browser scratch area; a calculator is optional unless rounding is requested.
- Supplied teaching case: For y=x², find the tangent line at x=2. Candidate answers: (A) y+4=4(x+2) (B) y=4x (C) y−4=2(x−2) (D) y−4=4(x−2).
- Independent check inputs: For y=x², find the tangent line at x=3. Candidate answers: (A) y+9=6(x+3) (B) y=6x (C) y−9=3(x−3) (D) y−9=6(x−3).
Learn the idea
A derivative measures instantaneous rate through a limiting difference quotient. Differentiation rules summarize that process for suitable functions, while a tangent line uses the derivative slope and the function’s actual point to model local behavior. A tangent line at an input passes through that function point and has derivative slope there. Point-slope form combines these two pieces into a local linear approximation.
Here is the method on the supplied teaching case. The point is (2,4) and derivative slope is 2(2)=4. Insert both into point-slope form. The tangent is a local linear model, not the whole parabola.
A correct slope alone does not locate the tangent line. Calculate the function output at the same input and include that point; the line generally differs from the curve at other inputs.
- Write or sketch the supplied teaching case, labeling its quantities and restrictions.
- Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
- Solve the independent check with its changed inputs before revealing feedback.
- Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
See the method in action
Teaching case: For y=x², find the tangent line at x=2. Candidate answers: (A) y+4=4(x+2) (B) y=4x (C) y−4=2(x−2) (D) y−4=4(x−2). Solution: The point is (2,4) and derivative slope is 2(2)=4. Insert both into point-slope form. The tangent is a local linear model, not the whole parabola. The final answer is y−4=4(x−2). A correct slope alone does not locate the tangent line. Calculate the function output at the same input and include that point; the line generally differs from the curve at other inputs.
Check your reasoning
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Helpful tips
- A tangent line at an input passes through that function point and has derivative slope there.
- A correct slope alone does not locate the tangent line.
See the answer and explanation
y−9=6(x−3)
The point is (3,9) and derivative slope is 2(3)=6. Insert both into point-slope form. The tangent is a local linear model, not the whole parabola.
Take it further
Changed-input task: For y=x², find the tangent line at x=9. Candidate answers: (A) y+81=18(x+9) (B) y=18x (C) y−81=18(x−9) (D) y−81=9(x−9). Show the mathematical reason, not only the selected answer. Expected result: y−81=18(x−9). The point is (9,81) and derivative slope is 2(9)=18. Insert both into point-slope form. The tangent is a local linear model, not the whole parabola.
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