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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 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}
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.
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
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