6.8 Summary (EMCHM)

• The limit of a function exists and is equal to \(L\) if the values of \(f(x)\) get closer to \(L\) from both sides as \(x\) gets closer to \(a\).

  \[\lim_{x\to a} f(x) = L\]

• Average gradient or average rate of change:

  \[\text{Average gradient } = \frac{f\left(x+h\right)-f\left(x\right)}{h}\]

• Gradient at a point or instantaneous rate of change:

  \[f'(x) = \lim_{h\to 0}\frac{f\left(x+h\right)-f\left(x\right)}{h}\]

• Notation

  \[{f}'\left(x\right)={y}'=\frac{dy}{dx}=\frac{df}{dx}=\frac{d}{dx}[f\left(x\right)]=Df\left(x\right)={D}_{x}y\]

• Differentiating from first principles:

  \[f'(x) = \lim_{h\to 0}\frac{f\left(x+h\right)-f\left(x\right)}{h}\]

• Rules for differentiation:

  • General rule for differentiation:

    \[\frac{d}{dx}\left[{x}^{n}\right]=n{x}^{n-1}, \text{ where } n \in \mathbb{R} \text{ and } n \ne 0.\]

  • The derivative of a constant is equal to zero.

    \[\frac{d}{dx}\left[k\right]= 0\]

  • The derivative of a constant multiplied by a function is equal to the constant multiplied by the derivative of the function.

    \[\frac{d}{dx}\left[k \cdot f\left(x\right) \right]=k \frac{d}{dx}\left[ f\left(x\right) \right]\]

  • The derivative of a sum is equal to the sum of the derivatives.

    \[\frac{d}{dx}\left[f\left(x\right)+g\left(x\right)\right]=\frac{d}{dx}\left[f\left(x\right) \right] + \frac{d}{dx}\left[g\left(x\right)\right]\]

  • The derivative of a difference is equal to the difference of the derivatives.

    \[\frac{d}{dx}\left[f\left(x\right) - g\left(x\right)\right]=\frac{d}{dx}\left[f\left(x\right) \right] - \frac{d}{dx}\left[g\left(x\right)\right]\]

• Second derivative:

  \[f''(x) = \frac{d}{dx}[f'(x)]\]

• Sketching graphs:

  The gradient of the curve and the tangent to the curve at stationary points is zero.

  Finding the stationary points: let \(f'(x) = 0\) and solve for \(x\).

  A stationary point can either be a local maximum, a local minimum or a point of inflection.

• Optimisation problems:

  Use the given information to formulate an expression that contains only one variable.

  Differentiate the expression, let the derivative equal zero and solve the equation.