Investigation: Sketching Graphs using Differential Calculus

Last updated about 5 years ago

11 Nsɛmmisa

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.

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

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).

Investigation: Sketching Graphs using Differential Calculus

Last updated about 5 years ago

11 Nsɛmmisa

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.

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

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

Use the stationary points to make a sign diagram.

For which values is f(x) increasing?

For which values is f(x) decreasing?

Write down the nature of each of the stationary points.

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

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.

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).

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

Summarise your investigation!

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!)

Investigation: Sketching Graphs using Differential Calculus

Investigation: Sketching Graphs using Differential Calculus

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

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

Use the stationary points to make a sign diagram.

For which values is f(x) increasing?

For which values is f(x) decreasing?

Write down the nature of each of the stationary points.

Find f"(x).

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

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

Summarise your investigation!

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!)

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

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

Use the stationary points to make a sign diagram.

For which values is f(x) increasing?

For which values is f(x) decreasing?

Write down the nature of each of the stationary points.

Find f"(x).

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

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

Summarise your investigation!

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!)

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

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