Unit 5 - Analytical Applications of Differentiation

Question 1

Mean Value Theorem

Answer 1

• Function f is continuous over an interval [a,b] and it’s differentiable over an interval (a,b) then there exists at least 1 value of c such that f’(c) = [f(b) - f(a)] / (b - a)
• Function f has to be differentiable

Question 2

Stationary points / Critical Points / Turning Points

Answer 2

At that point dy / dx = 0

Question 3

Interval where function is increasing / decreasing

Answer 3

1. Draw sign diagram for dy / dx
2. Look at the signs
3. If dy / dx > 0 –> function is increasing
4. If dy / dx < 0 –> function is decreasing

Question 4

Local / Relative maxima or minima

Answer 4

Peak turning point –> Local maxima
Low turning point –> Local minima
1. Using sign diagram of dy / dx
2. dy / dx changes from negative to positive = local minima
3. dy / dx changes from positive to negative = local maxima

Question 5

Global Maxima / Minima

Answer 5

1. Draw a table to find the y - coordinates for the corresponding x - coordinates of (endpoints, stationary points, vertical asymptotes)

Question 5

Intervals where f(x) is concave up / concave down

Answer 5

If:
d^2 y / dx > 0 –> concave up
d^2 x / dx < 0 –> concave down