Chapter 13: Properties of Curves

Question 1

Define a tangent to a curve, and find:
- Equation of a tangent at x= __
- Equation of a horizontal tangent
- Point where the tangent meets the line again
- Equations of possible tangents given an external point
- Common tangent

Answer 1

Tangent: Best approximating straight line to a curve at a given point

1) Equation of tangent at x=__
- Find f’(x) to find the gradient function
- Sub x=__ into f’(x) to find the gradient at the given point
- Find f(x) to find the y-coordinate at the given point
- Use the point-gradient form of the equation of a straight line to find the equation of the tangent

2) Equation of a horizontal tangent
- Find f’(x)
- Equate f’(x)=0 (horizontal line: gradient 0)
- Find x
- Find f(x) to find the y-coordinate AND the equation of the horizontal tangent

3) Point where the tangent meets the line again
- Find f’(x)
- Sub x of given point into f’(x) to find the gradient at given point
- Find equation of the tangent
- Equate the equation of the curve to the equation of the tangent
- Find unknown point

4) Possible equations of a tangent from a given external point
- Assign a general point
Eg. If y=x², then general point is (a,a²)
- Find f’(x)
- Find f’(a)
- Find the equation of a tangent in the general form
Eg. In terms of a
- Sub in coordinates of given point into the equation of a tangent to find unknown
- Sub in x=a to find actual equations of the tangents

5) Common tangent at (x,y)
- f’(x)=g’(x)
- f(x) = g(x)

Question 2

Define a normal to a curve and find the equation of a normal

Answer 2

Normal: Line perpendicular to a tangent at a given point (x=__)

Equation of a normal:
- Find f’(x)
- Sub in x to find gradient of the tangent
- Find gradient of the normal
(negative reciprocal of the gradient of the tangent)
- Sub in x into f(x) to find y-coordinate
- Use point-gradient form of equation of a straight line to find equation of the normal

Question 3

Explain how to determine if a function is increasing or decreasing

Answer 3

A function is increasing if:
- f(a)≤ f(b) when a<b
- f’(x) ≥ 0

A function is decreasing if:
- f(a)≥f(b) when a<b
- f’(x) ≤ 0

• Can be found by sketching the graph of the function, or by algebraically finding f’(x) and determining the range at which f’(x) ≥ 0 or f’(x) ≤ 0
  [using sign diagram]

Question 4

Define a stationary point and use sign diagrams to determine the nature of a stationary point

Answer 4

Stationary point:
- Graph changes from increasing to decreasing or decreasing to increasing
- f’(x) = 0

Sign diagram:
(+) to (-): local maximum point
(-) to (+): local minimum point
(+) to (+) OR (-) to (-): stationary point of inflection

Question 5

Use second derivatives (f”(x)) to find the shape of a function (f(x))
[concave up or concave down]

Answer 5

• If f”(x)≥0, the graph is concave up, as f’(x) is increasing (becoming more positive)
• If f”(x)≤0, the graph is concave down, as f’(x) is decreasing (becoming more negative)

Question 6

Define inflection points and determine if the inflection point is stationary or non-stationary

Answer 6

Inflection point:
- If f”(a) = 0
- If sign changes at f”(x)=0

Stationary point of inflection:
- If f’(a) = 0 (tangent at a is horizontal)

Non-stationary point of inflection:
- If f’(a)≠0 (tangent at a is not horizontal)