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Derivatives

11.2 Average Value, Straight Line Motion

Average Value of a Function

  • The average value of a function f(x) for an interval [a,b]
    is defined as:

\text { Average Value }=\frac{1}{b-a} \int_{a}^{b} f(x) d x

  • Graphically, it is equivalent as drawing a rectangle that has length b-a, and has an area that equals \int_{a}^{b} f(x) d x, or the area under the curve for the interval [a,b]. Therefore, the height of this rectangle is the average value of f(x)
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5.7 Maximum Rates of Increase or Decrease

Maximum Rates of Increase or Decrease

  • Earlier, we noticed how the first derivative f^{\prime}(x) can be used to find maximum and minimum values.
  • Similarly, we can find maximum and minimum values for f^{\prime}(x) by using its derivative, f^{{\prime}{\prime}}(x) i.e. the second derivative.
  • Recall that f^{\prime}(x) is interpreted as the rate of increase/decrease/change, therefore f^{{\prime}{\prime}}(x) is capable of finding the maximum and minimum rate of increase or decrease.
  • A table is constructed below to illustrate the cases.
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5.6 Maximum and Minimum in Real Life

Maximum and Minimum in Real Life

  • In real life, many things can be maximised or minimised (e.g. cost, area, volume, profit, sales, etc.) for various purposes (e.g. in manufacturing, marketing etc.).
  • Using differentiation and our knowledge in absolute maximum and minimum can help to solve these problems.

Example

A farmer has sufficient fencing to make a rectangular pen of perimeter 200 metres. What dimensions will give an enclosure of maximum area?

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5.5 Second Derivatives and Concavity

Second Derivatives and Point of Inflection

  • The first derivative of a function is known as f^{\prime}(x). Similarly, we denote the second derivative of a function as f^{{\prime}{\prime}}(x) or \frac{d^2y}{dx^2}.
  • If f^{{\prime}{\prime}}(x)>0 for an interval, the gradient of f(x) is increasing in the interval. The curve is said to be concave up (or have a ‘smiley face’)
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5.4 Maximum and Minimum Points

Local Maximum and Minimum Points

  • A local maximum point is when immediately to the left of the point, the gradient is positive, and immediately to right the gradient is negative. The table at the right summarises this.

Put simply, it is a point that has y-value larger than its surroundings.

  • A local minimum point is when immediately to the left of the point, the gradient is negative, and immediately to right the gradient is positive. The table at the right summarises this.
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5.3 First Derivative and Function Characteristics

Increasing and Decreasing Function

  • f(x) is strictly increasing in an interval (a,b) if f^{\prime}(x)>0 for all values of x in that interval.
    In this case, the tangent to any points in the interval has a positive gradient.
  • Similarly, f(x) is strictly decreasing in an interval if f^{\prime}(x)<0.
    The tangent has a negative gradient.
  • If f(x) is said to be strictly increasing (or decreasing) without any interval specified, then f^{\prime}(x)\geq0 (or f^{\prime}(x)\leq0)for ANY values of x (or precisely the domain of f(x)).
    Note: Notice the signs here are \geq, \leq instead of >, <.
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5.2 Rates of Change, Motion in a Straight Line

Finding Rates of Change

  • Recall that by definition, y=f^{\prime}(x) also represents the instantaneous rate of change of a function.
  • Also, by definition, the average rate of change across an interval [a,\ b] is given by

Average \: Rate \: of \: Change =\frac{f(b)-f(a)}{b-a}

  • Using differentiation, we can find the rate of change of a variable (e.g. volume, temperature, speed etc.) at any certain point of another variable (e.g. time).

Example

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5.1 Finding Tangents and Normals

Finding Tangents

  • Differentiation can help us find the equation of the tangent to a curve at a certain point.
  • To recall, to be able to draw a line, we would need either coordinates of two points, or coordinates of a point and the gradient. For the latter, say the line passes through point \left(x_{1},\ y_{1}\right) and has gradient m, then the line has equation

y-y_{1}=m(x-x_{1})

  • When you are asked to find the equation of a tangent to a curve, usually they would give you the coordinates of the point of interest.

In this case, you can use differentiation to find out the gradient of tangent line at that point, and together with the point of intersection, you can find out the line’s equation.

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4.3 Product and Quotient Rules

The Product Rule

  • Assume that there are functions f(x),\ g(x),\ h(x) where h(x)=f(x)\cdot g(x). Hence,

h^{\prime}(x)=f^{\prime}(x)\cdot g(x)+f(x)\cdot g^{\prime}(x)

  • Using Leibniz Notation, it is equivalent by saying

\frac{d y}{d x}=u \frac{d v}{d x}+v \frac{d u}{d x}

where u=g(x), v=f(x), y=uv

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4.2 Chain Rule

The Chain Rule

  • Assume that there are functions f(x),\ g(x),\ h(x) where h(x)=f(g(x)). Hence,

h^{\prime}(x)=f^{\prime}(g(x)) \cdot g^{\prime}(x)

i) If f(x)=c where c is a constant, then f^{\prime}(x)=0 (Constant function)

ii) If f(x)=k \cdot g(x), then f^{\prime}(x)=k \cdot g^{\prime}(x) (Multiple)

iii) If f(x)=g(x)+h(x), then f^{\prime}(x)=g^{\prime}(x)+h^{\prime}(x) (Sum)

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