Graphical gradients and area estimates · Higher
導入済み| English | 意味 |
|---|---|
| tangent gradient/ˈtændʒənt ˈɡreɪdɪənt/ | 接線の傾き |
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.
The method
- 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.
A chord and a tangent on d=t² both gave gradient 4 near t=2. Why can they differ in general?
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
The chord from t=1 to t=3 averages the changing rate; the tangent at t=2 matches the curve’s local direction. They coincide here only because the interval is symmetric about the contact point.
A worked case
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).
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(9-1)/(3-1)=4.
Where it goes wrong
- 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.
Not true: Area under any graph means the distance travelled. The area under a speed-time graph is distance; under other graphs the area means something else, or nothing useful.
Find the tangent gradient from (1,0) to (3,8).
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
(8-0)/(3-1)=4.
The area under a speed-time graph is distance.
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
True. The area under a speed-time graph is distance; under other graphs the area means something else, or nothing useful.
Applying it
- AQA A15 and R15 Higher require graph-based rates and area interpretation.
- This lesson uses graphical reasoning and geometric areas, not symbolic differentiation/integration.
Find the velocity trapezium displacement for 2 and 6 m/s over 3 s.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Average endpoint height 4 times width 3 gives 12.
Match each graph measure to its meaning.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Gradients measure rates and areas accumulate them; area under a distance-time graph has no such interpretation.
In your course
- AQA GCSE Mathematics (8300), Higher tier, specification section 3.2.
- Paper 1 is non-calculator: keep fractions, surds and π exact and show each step. On calculator papers, still write the expression before its value.
- Tangent gradient: The slope of a tangent, used to estimate a curve’s rate at one input.
- A chord gives an average rate between two inputs; a tangent estimates the local rate.
- Watch for: tangent estimates and curve chords use different pairs of points.