Investigation: Sketching Graphs using Differential Calculus
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Last updated over 4 years ago
11 questions
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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.
Question 1
1.
Consider f(x)={x^3}-{27x}
Write f'(x)
Question 2
2.
Using your answer in 1, find the coordinates of the stationary points of f(x)
Question 3
3.
Use the stationary points to make a sign diagram.
Question 4
4.
For which values is f(x) increasing?
Question 5
5.
For which values is f(x) decreasing?
Question 6
6.
Write down the nature of each of the stationary points.
Now you will need to explore the concavity of the graph of f(x).
Question 7
7.
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.
Question 8
8.
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).
Question 9
9.
Find the coordinates of any inflection points of f(x)
Question 10
10.
Summarise your investigation!
Question 11
11.
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!)
