Parametric first derivatives and vertical tangents
| English | 中文 | Pinyin |
|---|---|---|
| parametric gradient | 参数曲线斜率 | cān shù qū xiàn xié lǜ |
Two coordinates depend on the same slider. How do their separate rates combine into the curve’s slope?
- Two coordinates depend on the same slider. How do their separate rates combine into the curve’s slope?
- This lesson studies parametric gradient 参数曲线斜率: The ratio of the y-rate to the x-rate along a parametrised curve, when the x-rate is nonzero.
Choose the mathematical structure
- For x=x(t), y=y(t), the chain rule gives dy/dt=(dy/dx)(dx/dt), so dy/dx=(dy/dt)/(dx/dt) when dx/dt≠0. Differentiate the two coordinates separately, then divide in y-over-x order. The point is obtained from the same parameter value. A parameter need not represent physical time.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines parametric gradient?
The ratio of the y-rate to the x-rate along a parametrised curve, when the x-rate is nonzero.
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 x=t²+1, y=t³−3t, dx/dt=2t and dy/dt=3t²−3. At t=2 the point is (5,2) and gradient is 9/4, so the tangent is y−2=(9/4)(x−5). At t=1, the point (2,−2) has horizontal tangent y=−2 because dy/dt=0 while dx/dt=2. At t=0, (1,0) has vertical tangent x=1: dx/dt=0 but dy/dt=−3≠0. A regular local parametrisation then changes y while x has zero instantaneous change. For x=2cos t,y=3sin t, at t=π/4 the point is (√2,3√2/2) and slope is −3/2. At t=0 the ellipse tangent is vertical, x=2.
Parametric first derivatives and vertical tangents
For x=x(t), y=y(t), the chain rule gives dy/dt=(dy/dx)(dx/dt), so dy/dx=(dy/dt)/(dx/dt) when dx/dt≠0
Identify the appropriate changing variable and validate the derivative denominator or local branch.
For the cubic parametrisation at t=2, find dy/dx.
At t=2, dy/dt=9 and dx/dt=4, so the ratio is 9/4.
Test a tempting shortcut
- The ratio is dy/dt divided by dx/dt, not its reciprocal. If both rates vanish, 0/0 is inconclusive: x=t³,y=t³ still traces y=x with slope 1 through zero. If dx/dt=0 alone, inspect the local curve rather than report a finite numerical gradient. Repeated x values can correspond to different y values or branches; keep the stated parameter value.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
When both parametric coordinate rates are zero, the curve must have no tangent. This claim is false. Explain which definition or assumption it violates.
For the same curve at t=1, find dy/dx.
At t=1, the numerator is zero and denominator is 2, giving zero.
When both parametric coordinate rates are zero, the curve must have no tangent.
The ratio is dy/dt divided by dx/dt, not its reciprocal. If both rates vanish, 0/0 is inconclusive: x=t³,y=t³ still traces y=x with slope 1 through zero. If dx/dt=0 alone, inspect the local curve rather than report a finite numerical gradient. Repeated x values can correspond to different y values or branches; keep the stated parameter value.
Interpret a new situation
- Find the coordinate pair, calculate both first rates, then test whether the ratio is valid. State horizontal/vertical tangent lines separately when appropriate and preserve the parameter range. Check by eliminating the parameter when that gives a simple local relation. G5 requires first derivatives only; a second parametric derivative is outside this lesson’s stated scope.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
For the ellipse at t=π/4, find dy/dx.
(3cos t)/(−2sin t) at π/4 is −3/2.
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
- 7357 · A-level · G. 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 ratio of the y-rate to the x-rate along a parametrised curve, when the x-rate is nonzero. Choose the relationship, show the method, check its assumptions and interpret the result.