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# 数学代写|微积分代写Calculus代考|Average and Instantaneous Speed

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## 数学代写|微积分代写Calculus代考|Average and Instantaneous Speed

In the late sixteenth century, Galileo discovered that a solid object dropped from rest (initially not moving) near the surface of the earth and allowed to fall freely will fall a distance proportional to the square of the time it has been falling. This type of motion is called free fall. It assumes negligible air resistance to slow the object down, and that gravity is the only force acting on the falling object. If $y$ denotes the distance fallen in feet after $t$ seconds, then Galileo’s law is
$$y=16 t^2 \mathrm{ft}$$
where 16 is the (approximate) constant of proportionality. (If $y$ is measured in meters instead, then the constant is close to 4.9.)

More generally, suppose that a moving object has traveled distance $f(t)$ at time $t$. The object’s average speed during an interval of time $\left[t_1, t_2\right]$ is found by dividing the distance traveled $f\left(t_2\right)-f\left(t_1\right)$ by the time elapsed $t_2-t_1$. The unit of measure is length per unit time: kilometers per hour, feet (or meters) per second, or whatever is appropriate to the problem at hand.
Average Speed
When $f(t)$ measures the distance traveled at time $t$,
$$\text { Average speed over }\left[t_1, t_2\right]=\frac{\text { distance traveled }}{\text { elapsed time }}=\frac{f\left(t_2\right)-f\left(t_1\right)}{t_2-t_1}$$

## 数学代写|微积分代写Calculus代考|Average Rates of Change and Secant Lines

Given any function $y=f(x)$, we calculate the average rate of change of $y$ with respect to $x$ over the interval $\left[x_1, x_2\right]$ by dividing the change in the value of $y, \Delta y=f\left(x_2\right)-f\left(x_1\right)$, by the length $\Delta x=x_2-x_1=h$ of the interval over which the change occurs. (We use the symbol $h$ for $\Delta x$ to simplify the notation here and later on.)
DEFINITION The average rate of change of $y=f(x)$ with respect to $x$ over the interval $\left[x_1, x_2\right]$ is
$$\frac{\Delta y}{\Delta x}=\frac{f\left(x_2\right)-f\left(x_1\right)}{x_2-x_1}=\frac{f\left(x_1+h\right)-f\left(x_1\right)}{h}, \quad h \neq 0 .$$
Geometrically, the rate of change of $f$ over $\left[x_1, x_2\right]$ is the slope of the line through the points $P\left(x_1, f\left(x_1\right)\right)$ and $Q\left(x_2, f\left(x_2\right)\right)$ (Figure 2.1). In geometry, a line joining two points of a curve is called a secant line. Thus, the average rate of change of $f$ from $x_1$ to $x_2$ is identical with the slope of secant line $P Q$. As the point $Q$ approaches the point $P$ along the curve, the length $h$ of the interval over which the change occurs approaches zero. We will see that this procedure leads to the definition of the slope of a curve at a point.

Defining the Slope of a Curve
We know what is meant by the slope of a straight line, which tells us the rate at which it rises or falls – its rate of change as a linear function. But what is meant by the slope of a curve at a point $P$ on the curve? If there is a tangent line to the curve at $P$-a line that grazes the curve like the tangent line to a circle-it would be reasonable to identify the slope of the tangent line as the slope of the curve at $P$. We will see that, among all the lines that pass through the point $P$, the tangent line is the one that gives the best approximation to the curve at $P$. We need a precise way to specify the tangent line at a point on a curve.
Specifying a tangent line to a circle is straightforward. A line $L$ is tangent to a circle at a point $P$ if $L$ passes through $P$ and is perpendicular to the radius at $P$ (Figure 2.2). But what does it mean to say that a line $L$ is tangent to a more general curve at a point $P$ ?

To define tangency for general curves, we use an approach that analyzes the behavior of the secant lines that pass through $P$ and nearby points $Q$ as $Q$ moves toward $P$ along the curve (Figure 2.3). We start with what we can calculate, namely the slope of the secant line $P Q$. We then compute the limiting value of the secant line’s slope as $Q$ approaches $P$ along the curve. (We clarify the limit idea in the next section.) If the limit exists, we take it to be the slope of the curve at $P$ and define the tangent line to the curve at $P$ to be the line through $P$ with this slope.

The next example illustrates the geometric idea for finding the tangent line to a curve.

## 数学代写|微积分代写Calculus代考|Average and Instantaneous Speed

$$y=16 t^2 \mathrm{ft}$$

$$\text { Average speed over }\left[t_1, t_2\right]=\frac{\text { distance traveled }}{\text { elapsed time }}=\frac{f\left(t_2\right)-f\left(t_1\right)}{t_2-t_1}$$

## 数学代写|微积分代写Calculus代考|Average Rates of Change and Secant Lines

$$\frac{\Delta y}{\Delta x}=\frac{f\left(x_2\right)-f\left(x_1\right)}{x_2-x_1}=\frac{f\left(x_1+h\right)-f\left(x_1\right)}{h}, \quad h \neq 0 .$$

## MATLAB代写

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