## 数学代写|微积分代写Calculus代写|MTH263

2022年12月26日

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## 数学代写|微积分代写Calculus代写|The geometry of graphs

In Section $2.1$ we discussed the graph of a function $y=f(x)$ in terms of plotting points $(x, f(x))$ for many different values of $x$ and connecting the resulting points with straight lines. This is a standard procedure when using a computer and, if the function is well behaved and sufficiently many points are plotted, will produce a reasonable picture of the graph. However, as we noted at that time, this method assumes that the behavior of the graph between any two successive points is approximated well by a straight line. With a sufficient number of points and a differentiable function, this assumption will be reasonable. Yet to understand a graph fully, it is important to have alternative techniques to verify the picture at least qualitatively. We have already developed several important aids for understanding the shape of a graph, including techniques for determining the location of local extreme values and techniques for finding intervals where the function is increasing and intervals where it is decreasing. In this section we will use this information, along with additional information contained in the second derivative, to piece together a picture of the graph of a given function.
To see the importance of the second derivative, consider the graphs of $f(x)=x^2$ and $g(x)=\sqrt{x}$ on the interval $(0, \infty)$. Now
$$f^{\prime}(x)=2 x$$

and
$$g^{\prime}(x)=\frac{1}{2 \sqrt{x}},$$
so $f^{\prime}(x)>0$ and $g^{\prime}(x)>0$ for all $x$ in $(0, \infty)$. Thus $f$ and $g$ are both increasing on $(0, \infty)$. However, the graphs of $f$ and $g$, as shown in Figure 3.9.1, are dramatically different. The graph of $f$ is not only increasing, but is becoming steeper and steeper as $x$ increases, whereas the graph of $g$ is increasing, but flattening out as $x$ increases. In other words, $f^{\prime}$ is itself an increasing function, causing the rate of growth of the function to increase with $x$, while $g^{\prime}$ is a decreasing function, resulting in a decrease in the rate of growth of $g$ and a flattening out of the graph. In the terminology of the next definition, we say that the graph of $f$ is concave up on $(0, \infty)$ and the graph of $g$ is cuncave duwn un $(0, \infty)$.

Definition 3.9.1. Suppose $f$ is differentiable on the open interval $(a, b)$. If $f^{\prime}$ is an increasing function on $(a, b)$, then we say the graph of $f$ is concave up on $(a, b)$. If $f^{\prime}$ is a decreasing function on $(a, b)$, then we say the graph of $f$ is concave down on $(a, b)$.
Of course, to check for the intervals where $f^{\prime}$ is increasing and the intervals where $f^{\prime}$ is decreasing, we consider where $f^{\prime \prime}$, the derivative of $f^{\prime}$, is positive and where it is negative.

## 数学代写|微积分代写Calculus代写|The definite integral

As we discussed in Section 1.1, and mentioned again at the beginning of Section 3.1, there are two basic problems in calculus. In Chapter 3 we considered one of these, the problem of finding tangent lines to curves in the plane; we are now ready to turn to the second, quadrature, the problem of finding the area of a region in the plane. Although at first these problems would seem to have no connection, in Section $4.3$ we shall see that the Fundamental Theorem of Calculus relates them in an interesting and useful way. This theorem, first fully utilized by Newton and Leibniz, reveals that the problem of quadrature involves reversing the process of differentiation; as a consequence, the facility we developed in Chapter 3 for handling derivatives will be very helpful in many basic quadrature problems.

As illustrated in Figure 4.1.1, our basic example for studying quadrature will be the problem of finding the area of a region $R$ in the plane which is bounded above by the graph of a continuous function $f$ and below by an interval $[a, b]$ on the $x$-axis. Later we will see how to extend our techniques to more complicated planar regions. Recall that in Section $1.1$ we considered the problem of finding the area of the unit circle. In that case, we attacked the problem by approximating the area of the circle by the area of inscribed regular polygons, which were themselves divided into triangles. We used these to find the area of the circle by taking the limit of the areas of the inscribed polygons as the number of sides went to infinity. Here we will see that it is sufficient to use rectangles, rather than triangles, as our units of approximation. That is, we will approximate the area of the desired region by the area of rectangles and then ask how the approximation improves as we increase the number, while decreasing the width, of the approximating rectangles. We begin with an example.

Example 4.1.1. Consider the region $R$ beneath the graph of the function $f(x)=x^2+1$ and above the interval $[-1,2]$ on the $x$-axis. Let $A$ be the area of $R$. If $R_1$ is the rectangle with base on the interval $[-1,2]$ and height $f(2)=5$, then, since 5 is the maximum value of $f$ on $[-1,2], R_1$ contains $R$. We call $R_1$ a circumscribed rectangle for the region $R$. Hence the area of $R$ is less than the area of $R_1$, showing that $A \leq 15$. Similarly, if $R_2$ is the rectangle with base on the interval $[-1,2]$ and height $f(0)=1$, then, since 1 is the minimum value of $f$ on $[-1,2], R$ contains $R_2$. We call $R_2$ an inscribed rectangle for the region $R$. Hence the area of $R$ is greater than the area of $R_2$, showing that $A \geq 3$. See Figure 4.1.2.

# 微积分代考

## 数学代写|微积分代写Calculus代写|The geometry of graphs

$$f^{\prime}(x)=2 x$$

$$g^{\prime}(x)=\frac{1}{2 \sqrt{x}}$$

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## MATLAB代写

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