# 2.1 A preview of calculus  (Page 4/13)

 Page 4 / 13

## Estimation using rectangles

Estimate the area between the x -axis and the graph of $f\left(x\right)={x}^{2}+1$ over the interval $\left[0,3\right]$ by using the three rectangles shown in [link] . The area of the region under the curve of f ( x ) = x 2 + 1 can be estimated using rectangles.

The areas of the three rectangles are 1 unit 2 , 2 unit 2 , and 5 unit 2 . Using these rectangles, our area estimate is 8 unit 2 .

Estimate the area between the x -axis and the graph of $f\left(x\right)={x}^{2}+1$ over the interval $\left[0,3\right]$ by using the three rectangles shown here: 16 unit 2

## Other aspects of calculus

So far, we have studied functions of one variable only. Such functions can be represented visually using graphs in two dimensions; however, there is no good reason to restrict our investigation to two dimensions. Suppose, for example, that instead of determining the velocity of an object moving along a coordinate axis, we want to determine the velocity of a rock fired from a catapult at a given time, or of an airplane moving in three dimensions. We might want to graph real-value functions of two variables or determine volumes of solids of the type shown in [link] . These are only a few of the types of questions that can be asked and answered using multivariable calculus    . Informally, multivariable calculus can be characterized as the study of the calculus of functions of two or more variables. However, before exploring these and other ideas, we must first lay a foundation for the study of calculus in one variable by exploring the concept of a limit. We can use multivariable calculus to find the volume between a surface defined by a function of two variables and a plane.

## Key concepts

• Differential calculus arose from trying to solve the problem of determining the slope of a line tangent to a curve at a point. The slope of the tangent line indicates the rate of change of the function, also called the derivative . Calculating a derivative requires finding a limit.
• Integral calculus arose from trying to solve the problem of finding the area of a region between the graph of a function and the x -axis. We can approximate the area by dividing it into thin rectangles and summing the areas of these rectangles. This summation leads to the value of a function called the integral . The integral is also calculated by finding a limit and, in fact, is related to the derivative of a function.
• Multivariable calculus enables us to solve problems in three-dimensional space, including determining motion in space and finding volumes of solids.

## Key equations

• Slope of a Secant Line
${m}_{\text{sec}}=\frac{f\left(x\right)-f\left(a\right)}{x-a}$
• Average Velocity over Interval $\left[a,t\right]$
${v}_{\text{ave}}=\frac{s\left(t\right)-s\left(a\right)}{t-a}$

For the following exercises, points $P\left(1,2\right)$ and $Q\left(x,y\right)$ are on the graph of the function $f\left(x\right)={x}^{2}+1.$

[T] Complete the following table with the appropriate values: y -coordinate of Q , the point $Q\left(x,y\right),$ and the slope of the secant line passing through points P and Q . Round your answer to eight significant digits.

x y $Q\left(x,y\right)$ m sec
1.1 a. e. i.
1.01 b. f. j.
1.001 c. g. k.
1.0001 d. h. l.

a. 2.2100000; b. 2.0201000; c. 2.0020010; d. 2.0002000; e. (1.1000000, 2.2100000); f. (1.0100000, 2.0201000); g. (1.0010000, 2.0020010); h. (1.0001000, 2.0002000); i. 2.1000000; j. 2.0100000; k. 2.0010000; l. 2.0001000

A Function F(X)=Sinx+cosx is odd or even?
neither
David
Neither
Lovuyiso
f(x)=1/1+x^2 |=[-3,1]
apa itu?
fauzi
determine the area of the region enclosed by x²+y=1,2x-y+4=0
Hi
MP
Hi too
Vic
hello please anyone with calculus PDF should share
Which kind of pdf do you want bro?
Aftab
hi
Abdul
can I get calculus in pdf
Abdul
How to use it to slove fraction
Hello please can someone tell me the meaning of this group all about, yes I know is calculus group but yet nothing is showing up
Shodipo
You have downloaded the aplication Calculus Volume 1, tackling about lessons for (mostly) college freshmen, Calculus 1: Differential, and this group I think aims to let concerns and questions from students who want to clarify something about the subject. Well, this is what I guess so.
Jean
Im not in college but this will still help
nothing
how can we scatch a parabola graph
Ok
Endalkachew
how can I solve differentiation?
with the help of different formulas and Rules. we use formulas according to given condition or according to questions
CALCULUS
For example any questions...
CALCULUS
what is the procedures in solving number 1?
review of funtion role?
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find dy÷dx (y^2+2 sec)^2=4(x+1)^2
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anyone to explain some basic in calculus
I can
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