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3 years ago

Write an expression to represent each of the following sketches. Then give the value of each sketch.

Mathematics

1 answer:

zysi [14]3 years ago

4 0

The sketches are illustrations of inequalities

The first sketch is: $-2 \le x \le 5$ .
The second sketch is: $3 \ge x \ge -4$

Sketch A

In sketch A, we have the following observations:

The arrow faces left (this implies less than)
The dots on -2 and -5 are closed dots (this implies less than or equal to).

Hence, the inequality is:

$-2 \le x \le 5$

Sketch B

In sketch B, we have the following observations:

The arrow faces right (this implies greater than)
The dots on -4 and 3 are closed dots (this implies greater than or equal to).

Hence, the inequality is:

$3 \ge x \ge -4$

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Use the fundamental theorem of calculus to find the area of the region between the graph of the function x^5 + 8x^4 + 2x^2 + 5x

BaLLatris [955]

Answer:

The area of the region is 25,351 $units^2$ .

Step-by-step explanation:

The Fundamental Theorem of Calculus: if $f$ is a continuous function on $[a,b]$ , then

$\int_{a}^{b} f(x)dx = F(b) - F(a) = F(x) | {_a^b}$

where $F$ is an antiderivative of $f$ .

A function $F$ is an antiderivative of the function $f$ if

$F^{'}(x)=f(x)$

The theorem relates differential and integral calculus, and tells us how we can find the area under a curve using antidifferentiation.

To find the area of the region between the graph of the function $x^5 + 8x^4 + 2x^2 + 5x + 15$ and the x-axis on the interval [-6, 6] you must:

Apply the Fundamental Theorem of Calculus

$\int _{-6}^6(x^5+8x^4+2x^2+5x+15)dx$

$\mathrm{Apply\:the\:Sum\:Rule}:\quad \int f\left(x\right)\pm g\left(x\right)dx=\int f\left(x\right)dx\pm \int g\left(x\right)dx\\\\\int _{-6}^6x^5dx+\int _{-6}^68x^4dx+\int _{-6}^62x^2dx+\int _{-6}^65xdx+\int _{-6}^615dx$

$\int _{-6}^6x^5dx=0\\\\\int _{-6}^68x^4dx=\frac{124416}{5}\\\\\int _{-6}^62x^2dx=288\\\\\int _{-6}^65xdx=0\\\\\int _{-6}^615dx=180\\\\0+\frac{124416}{5}+288+0+18\\\\\frac{126756}{5}\approx 25351.2$

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natita [175]

Answer:

x = -5 ±4sqrt(3)

Step-by-step explanation:

(x + 5)^2 = 48

Take the square root of each side

sqrt((x + 5)^2) = ±sqrt(48)

x+5 = ±sqrt(16*3)

x+5 = ±4sqrt(3)

Subtract 5 from each side

x = -5 ±4sqrt(3)

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