91影视

Explain why terms in the sum in Example 6 with n equals to 4 are completely different from the terms in the sum when n equals to 3. How can the sum from $k=1\mathrm{to}k=4$ be smaller than the sum from $k=1tok=3$? What will happen as n gets larger in this example?

Fill in the blanks to complete each sum formula:

$\underset{k=1}{\overset{n}{鈭�}}k=\underline{}$

In the proof of the Fundamental Theorem of Calculus, the Mean Value Theorem is used to choose values $x_k^{*}$in each subinterval $\left[x_{k-1},x_k\right]$. Use the Mean Value Theorem in the same way to find the corresponding values $x_k^{*}$ for a Riemann sum approximation of ${鈭�}_0^4{x}^{2}dx$ with four rectangles.

Indefinite integrals: Use integration formulas, algebra, and

educated guess-and-check strategies to find the following

integrals .

$鈭�\frac{1-x+x^7}{x^2}dx$

Consider the function

$F\left(x\right)=\left\{\begin{array}{l}\ln \left|x\right|,ifx<0\\ \ln \left|x\right|+4,ifx>0\end{array}\right.$

Show that the derivative of this function is the function $f\left(x\right)=\frac{1}{x}$. Compare the graphs of $F\left(x\right)$ and $\ln \left|x\right|$, and discuss how this exercise relates to the second part of Theorem 4.16.

The definite integral of a function $f$on an interval $\left[a,b\right]$is defined as a limit of Riemann sums. How can it be that the sum of the areas of infinitely many rectangles that are each 鈥渋nfinitely thin鈥� is a finite number? On the one hand, shouldn鈥檛 it be infinite, since we are adding up infinitely many rectangles? On the other hand, shouldn鈥檛 it always be zero, since the width of each of the rectangles is approaching zero as $n鈫�鈭�$?

The following sum approximates the area between the graph of some function f and the x-axis from x = a to x = b. Do some 鈥渞everse engineering鈥� to determine the type of approximation (left sum, midpoint sum, etc.) and identify f(x), a, b, n, $鈭�x$, and $x_k$. Then sketch the approximation described.

$\underset{k=1}{\overset{100}{鈭�}}\frac{\sin \left(0.05\left(k-1\right)+\sin \left(0.05k\right)\right)}{2}\left(0.05\right)$

Are definite integrals the 鈥渋nverse鈥� of differentiation? In other words, does one undo the other? Simplify each of the following to answer this question:

$$\begin{gathered} \left(a\right){鈭�}_a^b{f}^{\text{'}}\left(x\right)dx \\ \left(b\right)\frac{d}{dx}{鈭�}_a^{b}f\left(x\right)dx \end{gathered}$$

State the Mean Value Theorem for Integrals, and explain what this theorem means. Include a picture with your explanation. What does the Mean Value Theorem for Integrals have to do with average values?

Fill in the blanks to complete each sum formula:

$\underset{k=1}{\overset{n}{鈭�}}{k}^3=\underline{}$