Derivatives Practice

Question 1

Answer 1

Question 2

Answer 2

Question 3

The following diagram shows part of the graph of the function f(x) = 2x². The line T is the tangent to the graph of f at x = 1.

1. Show that the equation of T is y = 4x – 2.
2. Find the x-intercept of T.

Answer 3

Question 4

Identify the two points of inflexion by looking at the graph.

Answer 4

Inflection is where smiley faces turn to frowny faces and vice-versa. This happens at B and D.

Question 5

Find f’(x).

Answer 5

Question 6

Answer 6

Question 7

Answer 7

Question 8

Answer 8

Question 9

Answer 9

Question 10

Write down the x-intercepts of the graph of the derivative function, f.

Answer 10

The x-intercepts of any graph are where y=0. So it is asking for the all the places in which f’(x)=0. We know that f’(x)=0 happens at local maxima and minima. Looking at the graph of f, I can see that this happens at x=-3, x=0, and x=2.

Question 11

Write down all values of x for which f’(x) is positive.

Answer 11

Question 12

At point D on the graph of f, the x-coordinate is –0.5. Explain why f′′(x) < 0 at D.

Answer 12

When we look at the point on the graph where the x-coordinate is -0.5, it occurs within a frowny face. This means at this point the function is concave down. And the intervals on which a function is concave down are the same ones in which f’‘(x) is negative. Therefore f’’(-0.5) < 0.

Question 13

Answer 13

If it’s a maximum, it must be at the top of a frowny face, which means the function is concave down at that point. We just check that f’‘(B) < 0.

Question 14

Answer 14

We know that A and B are critical/stationary points. This means the derivative of the function at these points has to be zero. Now you just solve f’(A)=0. (You could also solve f’(B)=0, but B looks more annoying.)