Log inSign up for FREE
Library

Investigation: Sketching Graphs using Differential Calculus

Last updated over 4 years ago
11 questions
For this investigation you will make an accurate sketch of the function f(x)={x^3}-{27x} without a GDC. In order to do this , you need to connect the ideas we have seen about the derivative.
1

Consider f(x)={x^3}-{27x}
Write f'(x)

1

Using your answer in 1, find the coordinates of the stationary points of f(x)

1

Use the stationary points to make a sign diagram.

1

For which values is f(x) increasing?

1

For which values is f(x) decreasing?

1

Write down the nature of each of the stationary points.

Now you will need to explore the concavity of the graph of f(x).
1

Find f"(x).

For any function f(x):
- If f"(x)<0 then the function is concave down at this point.
- If f"(x)>0 then the function is concave up at this point.
1

Evaluate f"(x) in each of the stationary points and determine the concavity of the function at each stationary point.

For any function f(x):
If f"(x) = 0 then x is an inflection point of the function (i.e. there is a change of concavity at this point).
1

Find the coordinates of any inflection points of f(x)

1

Summarise your investigation!

1

Finally, make a sketch of f(x)={x^3}-{27x}. Adapt your scale to your convenience (the x-axis and the y-axis do not need to have the same scale, but remember to be consistent in each of the axes!)