MTH6131 Chap.9 Tangent and Normal Lines and Related Rates
Tangent and Normal Lines and Related Rates
Two lines from one slope
The derivative at a point is the slope of the tangent there, so the tangent line is y − y₀ = f′(x₀)(x − x₀). The normal is the line through the same point at right angles to it, with slope −1/f′(x₀), and it is vertical when the tangent is horizontal.
The exercise set on this topic covers five families, and each has its own first move.
Five families of tangent problems
A tangent and normal at a given point need only the derivative there. Horizontal and vertical tangents of an implicit curve come from setting the numerator or the denominator of y′ to zero and substituting that condition back into the curve.
A tangent with a prescribed slope, parallel or perpendicular to a given line, sets the derivative equal to that slope and solves for the point. A tangent through a point off the curve names the unknown point of contact, writes the tangent there, and requires it to pass through the given point.
The angle between two curves where they cross comes from their two slopes through the tangent of the difference of angles, and the curves meet at right angles when the product of the slopes is −1.
Related rates
The schedule gives one lecture to related rates and differentials. A related-rates problem has quantities changing with time and an equation linking them at every instant.
Differentiating that equation with respect to time links their rates, and only then are the numbers for the particular instant substituted. The lecture example follows two ships on perpendicular courses and finds when they are closest by minimising the square of the distance between them.
Differentials
The differential dy = f′(x) dx is the change along the tangent line rather than along the curve.
For a small change in x it approximates the true change in y, which turns the tangent line into a quick estimator, and the direction of its error depends on whether the curve bends up or down.
What this chapter covers
- 01
Tangent and normal line equations
- 02
Horizontal and vertical tangents of implicit curves
- 03
Tangents with a prescribed slope
- 04
Tangents through a point off the curve
- 05
The angle between two curves
- 06
Related rates: differentiate, then substitute
- 07
Closest approach by minimising a squared distance
- 08
Differentials as tangent-line estimates
Worked example · free
The normal line to a square-root curve
- 2Find the point and the slope of the tangent.
- 2Write the tangent and then the normal, using the negative reciprocal slope.
Key terms
- Normal Line
- The line through a point of a curve perpendicular to the tangent there, with slope equal to minus the reciprocal of the tangent's slope.
- Point of Contact
- The point at which a tangent line touches a curve, which must be named as an unknown when the tangent passes through a point off the curve.
- Related Rates
- Problems in which quantities linked by an equation change with time, solved by differentiating the equation with respect to time.
- Differential
- The quantity dy = f′(x) dx, the change along the tangent line, used to approximate a small change in a function.
- Angle Between Curves
- The angle between the tangent lines of two curves at a point where they meet.
Tangent and Normal Lines and Related Rates FAQ
What is the difference between a tangent and a normal?
The tangent at a point has the curve's slope there; the normal passes through the same point at right angles to the tangent. If the tangent's slope is m, the normal's is minus one over m. When the tangent is horizontal, the normal is vertical and is written as x equals a constant.
How do I find a tangent line from a point that is not on the curve?
Do not put the given point into the derivative. Name the unknown point of contact as (a, f(a)), write the tangent there with slope f′(a), and require the line to pass through the given point. That gives an equation in a whose solutions are the points of contact, one tangent for each.
Why must I differentiate before substituting in a related-rates problem?
Because the numbers describe one instant, while the equation holds at every instant. Substituting first turns the variables into constants, and the derivative of a constant is zero, so the rates disappear. Differentiate the general relationship with respect to time, then substitute the values for the moment in question.
Exam move
For each of the five tangent families, keep one fully worked example as a template, and for related rates always draw the picture and label which quantities are changing before writing any equation. Most related-rates errors are errors in the picture, not in the calculus.
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