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11.3 Rate of Change, Total Change

Integration and Total Changes

  • In differentiation, we know that f^{\prime}(x) shows the rate of change of f(x). Therefore, by utilising integration, we can recover the original function from its rate of change function.
  • Essentially, from the concept of antiderivatives we have \int f^{\prime}(x) d x=f(x)+c.
  • Since f^{\prime}(x) represents rates of changes, thus integrating rates of changes basically gives the total change of a variable.
  • We have already seen this in 11.2 Average Value, Straight Line Motion, where we integrate the velocity (which is represents the rate of change of displacement) to find the total change in displacement.
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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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6.1 Antiderivatives of Functions

Antiderivatives of Specific Functions

  • We will be looking at antiderivatives of the polynomial, exponential, and trigonometric functions.
  • The formulas are as below:

i) Polynomial functions

\int x^{n} d x=\frac{1}{n+1} x^{n+1}+c \quad,\ n \neq-1

\int x^{-1} d x=\int \frac{1}{x} d x=\ln |x|+c

Note: Notice that in there is an absolute value over to make the result fit the domain for all logarithmic functions is (0,\ +\infty), and x=0 is not possible in the first place as it would make \frac{1}{x} undefined.

ii) Exponential functions

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2.2 Graphs of Antiderivative Functions

Deducing Antiderivative Function Graphs

Method 1

i) The sign of f^{\prime}(x) ( >0 or <0) decides the sloping (upward or downward) of f(x).

ii) Any x-intercept of f^{\prime}(x) (i.e. f^{\prime}(x)=0) shows the stationary point of f(x). If f^{\prime}(x) changes sign before and after passing the x-axis, then it is a turning point for f(x). If not, it is just a stationary point.

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