U2 - Curve Sketching

Question 1

What is the gradient of the NORMAL LINE relative to that of the TANGENT?

Answer 1

(-ve reciprocal)

-1/f’(a)

Question 2

How do we determine whether a function is INCREASING or DECREASING?

Answer 2

Increasing: slope of tangent @ specific point on an interval is POSITIVE

Decreasing: slope of tangent @ specific point on an interval is NEGATIVE.

Question 3

What is a CRITICAL POINT?

Answer 3

A point on function f(x) at which f’(x) = 0, or is UNDEFINED.

Question 4

What is a STATIONARY POINT?

Answer 4

A point on a function in which f’(x)=0. It could be a LOCAL MIN., LOCAL MAX, or STATIONARY INFLECTION.

Question 5

Differentiate between global extrema and local extrema.

Answer 5

Global: max or min value of y in the ENTIRE domain.

Local: max or min value, which is a turning point where f’(x)=0 (curve increases then decreases, or vice versa)

Question 6

What is a STATIONARY POINT OF INFLECTION?

Answer 6

A point on f(x) at which the curve changes shape or curvature. If f’(x)=0 at the critical point x=c, then this could indicate a local max, min, or stationary point of inflection.

Question 7

To determine the global maxima or minima of a function on an interval…

Answer 7

You must check the value of the function at the end points!

Ex. Given f(x) on the interval -2 less than or equal to x, which is less than of equal to 5

Question 8

The _____________ test can be used to find intervals of increase/decrease and stationary points of f(x), whereas the _____________ test can be used to find concavity and inflexion in f(x).

Answer 8

1st derivative test

2nd derivative test

Question 9

How do we determine the CONCAVITY of a function?

Answer 9

2nd derivatives test

1. Find f’’(x)
2. Find x values
3. Make interval line, plot the x values on the line, and substitue values in to determine concavity.

Concave DOWN if f’’(a) < (or equal to) 0
Concave UP if f’’(a) > (or equal to) 0