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2 years ago

Graph y<2/3x+1 Click or tap the graph to plot a point.

Mathematics

1 answer:

vivado [14]2 years ago

5 0

Answer:

See below

Step-by-step explanation:

a) Create a table containing a few values of x and y

x y

-2 -⅓

-1 ⅓

0 1

1 1⅔

2 2⅓

(b) Draw the graph

Plot the points and draw a straight line through them. Make it a dashed line, because the line does not include the points.

Your graph should look like Fig. 1.

(d) Check a point to see if it satisfies the inequality.

An easy one to check is the origin, (0, 0).

0 < ⅔(0) + 1

0 < 1

The area below the graph satisfies the inequality.

(e) Shade in the area below the graph

This shows that all points in that area satisfy the inequality. The final graph should look like Figure 2.

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Find the length of the following two-dimensional curve. r (t ) = (1/2 t^2, 1/3(2t+1)^3/2) for 0 < t < 16

andrezito [222]

Answer:

r = 144 units

Step-by-step explanation:

The given curve corresponds to a parametric function in which the Cartesian coordinates are written in terms of a parameter "t". In that sense, any change in x can also change in y owing to this direct relationship with "t". To find the length of the curve is useful the following expression;

$r(t)=\int\limits^a_b ({r`)^2 \, dt =\int\limits^b_a \sqrt{((\frac{dx}{dt} )^2 +\frac{dy}{dt} )^2)} dt$

In agreement with the given data from the exercise, the length of the curve is found in between two points, namely 0 < t < 16. In that case a=0 and b=16. The concept of the integral involves the sum of different areas at between the interval points, although this technique is powerful, it would be more convenient to use the integral notation written above.

Substituting the terms of the equation and the derivative of r´, as follows,

$r(t)= \int\limits^b_a \sqrt{((\frac{d((1/2)t^2)}{dt} )^2 +\frac{d((1/3)(2t+1)^{3/2})}{dt} )^2)} dt$

Doing the operations inside of the brackets the derivatives are:

1 ) $(\frac{d((1/2)t^2)}{dt} )^2= t^2$

2) $\frac{(d(1/3)(2t+1)^{3/2})}{dt} )^2=2t+1$

Entering these values of the integral is

$r(t)= \int\limits^{16}_{0} \sqrt{t^2 +2t+1} dt$

It is possible to factorize the quadratic function and the integral can reduced as,

$r(t)= \int\limits^{16}_{0} (t+1) dt= \frac{t^2}{2} + t$

Thus, evaluate from 0 to 16

$\frac{16^2}{2} + 16$

The value is $r= 144 units$

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3 years ago

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