Tangent and normal equations and gradient conditions
도입됨| English | 의미 |
|---|---|
| normal line/ˈnɔːml laɪn/ | 법선 |
A road’s tangent gives its local direction; a cross-road follows the normal. Why must both lines use the same curve point?
- A road’s tangent gives its local direction; a cross-road follows the normal. Why must both lines use the same curve point?
- This lesson studies normal line 法线: A line through the curve point perpendicular to its local tangent.
Choose the mathematical structure
- At a differentiable point (a,f(a)), the tangent gradient is m=f′(a) and its equation is y−f(a)=m(x−a). For a finite nonzero m, the normal gradient is −1/m. Find points satisfying a given gradient condition by solving for f′(x), then evaluate the original f. Parallel lines have equal gradients; finite nonzero perpendicular gradients multiply to −1.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
At x=1 on f=x³-3x the tangent is horizontal. What is the normal?
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
The normal is perpendicular to the tangent; a horizontal tangent (m=0) forces a vertical normal, stated as x=1. The formula -1/m applies only to finite nonzero m.
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 f=x³−3x at x=2, f(2)=2 and f′(2)=9. The tangent is y−2=9(x−2), or y=9x−16. The normal is y−2=−(1/9)(x−2). Tangents parallel to y=9x+4 satisfy 3x²−3=9, giving x=±2: points (2,2) and (−2,−2). The second tangent is y+2=9(x+2). At x=1 the tangent is horizontal y=−2 and the normal is vertical x=1; no finite negative reciprocal of zero exists. For the circle at (5,0), the tangent is vertical x=5 and the normal horizontal y=0. The secant through (−2,−2) and (2,2) has gradient 1, unlike either local tangent gradient 9.
Tangent and normal equations and gradient conditions
At a differentiable point (a,f(a)), the tangent gradient is m=f′(a) and its equation is y−f(a)=m(x−a)
Connect derivative calculations to their original point, domain and stated rate law.
For f=x³−3x, find the tangent gradient at x=2.
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
f′=3x²−3, giving 9 at x=2.
Test a tempting shortcut
- The derivative supplies a gradient, not the y-coordinate. Use the original curve for the point. The normal’s reciprocal includes a minus sign and is unavailable for a horizontal tangent; give a vertical line instead. A tangent can meet the curve again elsewhere. A given line’s intercept does not affect a parallel-gradient condition, but both tangent and normal must pass through the stated curve point.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
A normal to a horizontal tangent has finite gradient zero. This claim is false. Explain which definition or assumption it violates.
Find the normal gradient there.
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
For the finite nonzero tangent gradient 9, the normal gradient is −1/9.
A normal to a horizontal tangent has finite gradient zero.
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
The derivative supplies a gradient, not the y-coordinate. Use the original curve for the point. The normal’s reciprocal includes a minus sign and is unavailable for a horizontal tangent; give a vertical line instead. A tangent can meet the curve again elsewhere. A given line’s intercept does not affect a parallel-gradient condition, but both tangent and normal must pass through the stated curve point.
Interpret a new situation
- Find the point and gradient separately, then use point-slope form. If a gradient condition gives several inputs, retain every allowed point and line. State horizontal/vertical cases without inventing an infinite numerical gradient. Verify the line passes through the point and that a finite nonzero tangent/normal gradient pair multiplies to −1.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
How many tangents to f are parallel to y=9x+4?
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
3x²−3=9 has the two allowed inputs ±2.
Match each line at (a,f(a)) to its equation or gradient.
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
The derivative supplies the gradient and the ORIGINAL curve supplies the point; the two must be evaluated separately at the same x.
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.
A line through the curve point perpendicular to its local tangent. Choose the relationship, show the method, check its assumptions and interpret the result.