Graphical gradients and area estimates · Higher
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| tangent gradient/ˈtændʒənt ˈɡreɪdɪənt/ | 切线斜率 | qiè xiàn xié lǜ |
Which graphical quantity answers the question?
- A vehicle speeds up along a curve. Is the slope between its start and finish enough to describe its speed halfway?
- This lesson studies tangent gradient 切线斜率: The slope of a tangent, used to estimate a curve’s rate at one input.
Choose the mathematical structure
- A chord gives an average rate between two inputs; a tangent estimates the local rate. Read two well-separated points on the drawn tangent to reduce measurement error. On a velocity-time graph, area represents displacement. Split it into triangles/trapezia or estimate curved area using narrow strips.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines tangent gradient?
The slope of a tangent, used to estimate a curve’s rate at one input.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
For a distance curve d=t², the chord from (1,1) to (3,9) has gradient (9-1)/(3-1)=4. A tangent at (2,4) passes through (1,0) and (3,8), giving gradient 4 without calculus. A velocity-time trapezium with endpoint velocities 2 and 6 m/s over 3 s has area (2+6)×3/2=12 m. Smaller strips can improve a curved-area estimate; curvature affects over/underestimation. A cost-versus-quantity tangent through (2,12) and (6,28) gives a local rate (28-12)/(6-2)=4 yuan per extra item, linking graph slope to a financial interpretation.
Graphical gradients and area estimates
A chord gives an average rate between two inputs; a tangent estimates the local rate
Compare the model with the worked case and explain one change.
Find the chord gradient from (1,1) to (3,9).
(9-1)/(3-1)=4.
Test a tempting shortcut
- Tangent estimates and curve chords use different pairs of points. Area under a distance-time graph is not distance travelled. A strip estimate is approximate unless the graph is linear on each strip.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Area under any graph always means the distance travelled. This claim is false. Explain which definition or assumption it violates.
Find the tangent gradient from (1,0) to (3,8).
(8-0)/(3-1)=4.
Area under any graph always means the distance travelled.
Tangent estimates and curve chords use different pairs of points. Area under a distance-time graph is not distance travelled. A strip estimate is approximate unless the graph is linear on each strip.
Interpret a new situation
- AQA A15 and R15 Higher require graph-based rates and area interpretation. This lesson uses graphical reasoning and geometric areas, not symbolic differentiation/integration.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the velocity trapezium displacement for 2 and 6 m/s over 3 s.
Average endpoint height 4 times width 3 gives 12.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 8300 · Higher · 3.2. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The slope of a tangent, used to estimate a curve’s rate at one input. Choose the relationship, show the method, check its assumptions and interpret the result.