Introduction to Derivatives (DP IB Applications & Interpretation (AI): HL): Revision Note

Introduction to derivatives

What is a limit?

• The limit of a function is the value a function (of $x$) approaches as $x$ approaches a particular value from either side

  • Limits are of interest when the function is undefined at a particular value

  • For example, the function$f\left(x\right)=\frac{x^4-1}{x-1}$ will approach a limit (the value 4) as $x$ approaches 1 from both below and above

    • But the function is undefined at $x=1$ as this would involve dividing by zero

What might I be asked about limits?

• You may be asked to predict or estimate limits from a table of function values or from the graph of $y=f\left(x\right)$

• You may be asked to use your GDC to plot the graph and use values from it to estimate a limit

What is a derivative?

• Calculus is about rates of change

  • the way a car’s position on a road changes is its speed (velocity)

  • the way the car’s speed changes is its acceleration

• The gradient (rate of change) of a non-linear function varies with $x$

• The derivative of a function is a function that gives the gradient at each value of $x$

• The derivative is also called the gradient function

How are limits and derivatives linked?

• Consider the point $P$ on the graph of $y=f\left(x\right)$ as shown below

  • $\left[P{Q}_i\right]=P{Q}_1,P{Q}_2,...$ is a series of chords

• The gradient of the function$f\left(x\right)$ at the point $P$ is equal to the gradient of the tangent at point $P$

• The gradient of the tangent at point $P$ is the limit of the gradient of the chords $\left[P{Q}_i\right]$ as point $Q$ ‘slides’ down the curve and gets ever closer to point $P$

• The gradient of the function changes as $x$ changes

• The derivative is the function that calculates the gradient as an output if you input a value of $x$

What is the notation for derivatives?

• For the function $y=f\left(x\right)$, the derivative, with respect to $x$, is written as

$\frac{\mathrm{d}y}{\mathrm{d}x}=f^{\text{'}}\left(x\right)$

• Different variables may be used

  • e.g. If $V=f\left(s\right)$ then  $\frac{\mathrm{d}V}{\mathrm{d}s}=f^{\text{'}}\left(s\right)$