91影视

In Examples 1 and 4 we found four different approximations

for the area between the graph of f (x) = x² 鈭�2x+2 and the x-axis on [1, 3].

(a) Based on the pictures of these approximations given in the reading, which are over-approximations? Which are under-approximations? Which approximation looks like it might be closest to the actual area

under the curve?

(b) The actual area (which we won鈥檛 know how to calculate until Section 4.5) under the graph of f (x) = x² 鈭� 2x + 2 on [1, 3] is equal to $\frac{14}{3}$鈮� 4.6667. Given

this information, which of the approximations in Examples 1 and 4 was the most accurate? Was that what you expected?

Determine the independent variable of the function$A\left(x\right)={鈭�}_a^{x}f\left(t\right)dt$

what the dependent variable represents and to explain for calling to be the dummy variable. In the above expression, x is the independent variable.

Give precise mathematical definitions or descriptions of each of the concepts that follow. Then illustrate the definition with a graph or algebraic example, if possible.

• the definite integral of a function f on an interval $[a,b]$as a limit of Riemann sums, including the definitions of $螖x$, $x_k$, and $x_k^{*}$
  

Notation: Describe the meanings of each of the following mathematical expressions or how they are commonly used in this chapter:

$[x_{k鈭�1},x_k]$

Calculating sums: Determine the value of each of the sums that

follow. Some can be computed directly, some require the use

of sum formulas, and for some you will have to also compute

a limit .

$\underset{n鈫�鈭�}{\lim }\underset{k=1}{\overset{n}{鈭�}}\frac{k^2+1}{n^3}$

Approximate the same area as earlier but this time with eight rectangles is this over approximation or under approximation of the exact area under the graph

Find a function f that has the given derivative f and value f(c), if possible.

$f\text{'}\left(x\right)=\frac{5}{3x-2},f\left(1\right)=8$

What important theorem is the key to proving the Fundamental Theorem of Calculus?

Use a sentence to describe what the notation $\underset{k=3}{\overset{87}{鈭�}}{k}^2$means. (Hint: Start with 鈥淭he sum of....鈥�)

Suppose f is positive on (鈭掆垶, 鈭�1] and [2,鈭�) and negative on the interval [鈭�1, 2]. Write (a) the signed area and (b) the absolute area between the graph of f and the x-axis on [鈭�3, 4] in terms of definite integrals that do not involve absolute values.