Derivatives & Graphs (DP IB Analysis & Approaches (AA): SL): Revision Note

Derivatives & graphs

How are derivatives and graphs connected?

• If the graph of a function$y=f\left(x\right)$ is known, or can be sketched, then it is also possible to sketch the graphs of the first and second derivatives$y=f^{\text{'}}\left(x\right)$ and$y=f^{\text{'}\text{'}}\left(x\right)$

• The key properties of a graph include

  • the$\mathit{y}$-axis intercept

  • the$\mathit{x}$-axis intercepts – the roots of the function, where$f\left(x\right)=0$

  • stationary points, where$f^{\text{'}}\left(x\right)=0$

    • turning points – (local) minimum and maximum points

    • (horizontal) points of inflection

  • (non-stationary,$\mathrm{f}\text{'}\left(x\right)\ne 0$) points of inflection

  • asymptotes – vertical and horizontal

  • intervals where the graph is increasing and decreasing

  • intervals where the graph is concave down and concave up

• Not all graphs have all of these properties and not all properties can be determined without knowing the expression of the function explicitly

• However questions will provide enough information to sketch

  • the shape of the graph

  • some of the key properties such as roots or turning points

How do I sketch the graph of y = f'(x) from the graph of y = f(x)?

• You can sketch many of the most important features of $y=f^{\text{'}}\left(x\right)$ directly from the graph of $y=f\left(x\right)$

How do I sketch the graph of y = f''(x) from the graph of y = f(x)?

• First sketch the graph of$y=f^{\text{'}}\left(x\right)$ from$y=f\left(x\right)$, as per the above process

  • Then, using the same process, sketch the graph of$y=f^{\text{'}\text{'}}\left(x\right)$ from the graph of$y=f^{\text{'}}\left(x\right)$

• There are a couple of things you can deduce about the graph of$y=f^{\text{'}\text{'}}\left(x\right)$ directly from the graph of$y=f\left(x\right)$

Is it possible to sketch the graph of y = f(x) from the graph of a derivative?

• It is possible to sketch a graph of$y=f\left(x\right)$ from a graph of $y=f^{\text{'}}\left(x\right)$ by considering the reverse of the above

• There are some properties of the graph of$y=f\left(x\right)$ that cannot be determined from the graph of$y=f^{\text{'}}\left(x\right)$

  • the$\mathit{y}$-axis intercept

  • the intervals for which$y=f\left(x\right)$ is positive and negative

  • the roots of$y=f\left(x\right)$

• Unless a specific point the curve passes through is known, the constant of integration cannot be determined

  • Therefore the exact location of the curve will remain unknown

    • but it will still be possible to sketch its shape

• If starting from the graph of the second derivative,$y=f^{\text{'}\text{'}}\left(x\right)$, it is easier to sketch the graph of$y=f^{\text{'}}\left(x\right)$ first, then sketch$y=f\left(x\right)$