How do a point and a derivative determine a tangent?
You will be able to: Write a tangent-line equation using a point and its derivative.
How do a point and a derivative determine a tangent?
A straight ruler placed along a curve near one point follows its local direction. To describe that line, you need the point it passes through and its slope.
A useful starting point: How do local slopes become a new function? →
Words and symbols before equations
- Tangent line
- Line through (a,f(a)) with slope f′(a), when the derivative is finite.
- Point-slope form
- y−y₀=m(x−x₀), the line through (x₀,y₀) with slope m.
- Horizontal tangent
- A tangent with slope zero.
- Local
- Near a specified input, not necessarily across the entire graph.
What this picture assumes
Original equation-driven model; readouts are rounded. Graphs have labeled linear scales and finite sampled windows; algebra supplies exact conclusions. f=x². The tangent is y=a²+2a(x−a). Independent horizontal and vertical scales; use coordinates, not screen angle.
Read the picture in three steps
- Read the axes and labels first. Identify what each symbol and line represents. Read the units and fixed conditions before comparing quantities.
- a=1; f(a)=1; f′(a)=2. Tangent: y−1=2(x−(1)). Blue: f=x²; orange: tangent. The second blue graph is f′=2x. All coordinates are dimensionless; graph heights and rates have distinct roles.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
For f(x)=x² the derivative is 2x. At a=1 the point is (1,1) and the slope is 2, so y−1=2(x−1).
The derivative supplies only the slope. The original function supplies the vertical coordinate; using (a,f′(a)) would usually anchor the wrong line.
A tangent can cross a curve. For x³ at zero the tangent y=0 crosses the curve; touching without crossing is not a general definition.
Read numeric axes rather than screen angle. A stretched graph can make the same slope look different, and the tangent is a line rather than the full curve.
A worked example, step by step
Find the tangent to f(x)=x² at x=−1.
- Compute the point: f(−1)=1.
- Compute the derivative rule f′(x)=2x.
- Evaluate slope f′(−1)=−2.
- Write y−1=−2(x+1), or y=−2x−1; at x=−1 it returns 1.
Do not use the derivative value as the tangent’s vertical coordinate.
Must a tangent intersect the curve exactly once?
Compare with an explanation
No. The definition uses local slope at the contact point; other intersections can occur.
Predict. Change one thing. Explain.
Move the contact input. Predict where the tangent becomes horizontal and check that the tangent always passes through the marked function point.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
a=1; f(a)=1; f′(a)=2. Tangent: y−1=2(x−(1)). Blue: f=x²; orange: tangent. The second blue graph is f′=2x. All coordinates are dimensionless; graph heights and rates have distinct roles.
Original equation-driven model; readouts are rounded. Graphs have labeled linear scales and finite sampled windows; algebra supplies exact conclusions. f=x². The tangent is y=a²+2a(x−a). Independent horizontal and vertical scales; use coordinates, not screen angle.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using a difference quotient, tangent slope, or derivative rule with its domain conditions. Identify what the representation cannot tell you.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionWrite the tangent to y=x² at x=3 and verify its slope and contact point.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: The point is (3,9).
- 1 point: The derivative is 2x, so the slope is 6.
- 1 point: The line is y−9=6(x−3), or y=6x−9.
- 1 point: At x=3 it gives y=9 and its constant slope is 6.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1Which rule supplies the contact height?
The original f(a).
RECALL 2Which rule supplies the slope?
The derivative f′(a).
RECALL 3Can a tangent cross its curve?
Yes; for example y=0 is tangent to x³ at zero.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How do a point and a derivative determine a tangent?
- Tangent: y−f(a)=f′(a)(x−a).
- A finite derivative is the tangent slope.
Remember: Do not use the derivative value as the tangent’s vertical coordinate.
Conditions: Original equation-driven model; readouts are rounded. Graphs have labeled linear scales and finite sampled windows; algebra supplies exact conclusions. f=x². The tangent is y=a²+2a(x−a). Independent horizontal and vertical scales; use coordinates, not screen angle.
Refresh Kid · AP Calculus AB Unit 2 · Objectives CHA-2.B, CHA-2.C · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 2.2, CHA-2.B, CHA-2.C. CED effective Fall 2020, with Fall 2026 clarifications, checked September 17, 2026. Unit 2 has 10 official topics. Focused lesson titles and questions are original Refresh Kid teaching material.
Derivative definitions require finite two-sided limits at interior inputs. Graphs and finite tables supply evidence and estimates, not universal proofs. Continuity is necessary but insufficient for differentiability. Rules retain their original domain restrictions, and trigonometric inputs use radians. Chain-rule compositions, implicit differentiation, inverse-function differentiation and L’Hôpital’s rule are deferred to later units. Piecewise joins are checked for continuity before their side slopes are compared.
The Organic Chemistry Tutor video titles, creators and descriptions were checked; full videos were not reviewed. Khan Academy’s unit destination was checked; its JavaScript lesson content was not fully readable by the research tool. OpenStax Sections 3.1, 3.3 and 3.5 were consulted for conceptual cross-checking. No creator scripts, questions, diagrams or artwork were copied. Refresh Kid is not endorsed by these providers.
GitHub’s 3D website collection and its camera-control example informed optional spatial inspection. Graph geometry is generated from the stated original equations; self-hosted Three.js retains its MIT license. The optional 3D model uses separate planes for f and f′. Depth distinguishes panels, not an additional variable; camera rotation does not change their values. Use the complete labeled 2D graphs and text to read precise coordinates and slopes.
Independent teacher review and observation of students remain pending. Checks do not certify scientific accuracy, accessibility or learning effectiveness. This is a review edition.
Optional official resource: Released AP Calculus AB questions and scoring guides. The archive spans multiple units; no entire exam question is assigned to this lesson.
Learn → Explore → Practice → Review is informed by the IES learning guide; this exact implementation has not been evaluated with learners.
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