Chapter 10 Differentiation

Question 1

How do you determine the nature of stationary points?

Answer 1

The maximum and minimum are determined by the sign of the gradient on either side of the stationary point.

Question 2

How can you determine if f(x) is increasing on the interval (a, b)?

Answer 2

f(x) is increasing for a < x < b if f’(x) > 0 for a < x < b.

Question 3

How can you determine if f(x) is decreasing on the interval (a, b)?

Answer 3

f(x) is decreasing for a < x < b if f’(x) < 0 for a < x < b.

Question 4

What is the gradient of the tangent at a point (x, y)?

Answer 4

The gradient of the tangent is m = f’(x) and is the same as the gradient of the curve at that point.

Question 5

What is the equation of the tangent line at (x, y)?

Answer 5

The equation of the tangent is y - y₁ = m(x - x₁).

Question 6

What is the gradient of the normal line at a point (x, y)?

Answer 6

The gradient of the normal is m’ = -1/m and is the perpendicular gradient to the tangent and the curve

Question 7

What is the equation of the normal line at (x, y)?

Answer 7

The equation of the normal is y - y₁ = m’(x - x₁).

Question 8

How is the derivative of y with respect to x denoted?

Answer 8

The rate of change of y with respect to x is called the derived function or derivative, denoted as dy/dx.

Question 9

What is the notation for the derivative of f(x)?

Answer 9

In function notation, the derivative of f(x) is written as f’(x).

Question 10

What is the formula for differentiation from first principles?

Answer 10

This expression is used to find the derivative of a function.

Question 11

What does it indicate if the second derivative d²y/dx² is greater than 0?

Answer 11

The point is a minimum.

This means the function is concave up at that point.

Question 12

What does it indicate if the second derivative d²y/dx² is less than 0?

Answer 12

The point is a maximum.

This means the function is concave down at that point.

Question 13

What is the second derivative written as?

Answer 13

d²y/dx² or f ‘‘(x)

It represents the rate of change of the gradient from the first derivative.

Question 14

What are the basic differentiation rules?

Answer 14