2.6: Summary
- Page ID
- 121091
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- Graphs of time-dependent data are helpful to visualize trends such as increasing and decreasing values, steepness, linearity, and so on.
- An average rate of change is the ratio of change in a dependent variable \((y)\) over a range of the independent variable \((x)\), often denoted \(\Delta y / \Delta x\).
- A secant line is a straight line through any two points on the graph of a function. The slope of a secant line is the average rate of change of the function over interval between the \(x\) coordinates of the two points.
- The average rate of change of a function \(y=f(x)\) on \(x_{0} \leq x \leq x_{0}+h\) can be computed by the ratio
\[\frac{\Delta y}{\Delta x}=\frac{\text { Change in } y}{\text { Change in } x}=\frac{\left[f\left(x_{0}+h\right)-f\left(x_{0}\right)\right]}{h} \nonumber \]
- The instantaneous rate of change of a function \(f(x)\) at \(x_{0}\) can be found by taking the limit as \(h \rightarrow 0\) of the average rate of change on \(x_{0} \leq x \leq x_{0}+h\).
- The derivative of a function \(y=f(x)\) at a point \(x_{0}\) is the same as the instantaneous rate of change of \(f\) at \(x_{0}\).
- In the case of time-dependent data, refining the data can lead to a better and better approximation of instantaneous rates of change.
- This chapter explored data related to the following time-dependent processes: (a) height of a falling object; (b) temperature of heating/cooling milk; and (c) swimming velocity of Bluefin tuna.
- How do we calculate average rate of change of a time dependent process over a given interval?
- Over what interval does the function depicted in the graph below have the greatest average rate of change? Smallest average rate of change?
- Given the function defined by \(\{(1,3),(2,5),(3,7)\}\), how many different secant lines can be formed?
- Use the definition to calculate the derivative of \(f(x)=4 x^{2}+3\) at \(x_{0}=1\).