Answer:
(0, 7)
Step-by-step explanation:
The point that crosses the y-axis is called the y-intercept of the line.
We are given an equation of the line and we need to determine the y-intercept. This can be done by determining the constant in the equation.
What are constants?
Constants are such terms that include only numbers. There should be no variables in constants.
<u>We know that:</u>
The constant we can see in the equation is "7". Therefore, the y-intercept is 7.
Note: The y-intercept of the line must have an x-coordinate of 0.
<u>Therefore,</u>
⇒ Coordinates of y-intercept: (x, y) ==> (0, 7).
You can combine the -6 and -1 to -7. So the expression can be simplified as -3x-7
Subtract the y values to one side then the numbers to the other so as x values to one side then the numbers to the other
(a) By the fundamental theorem of calculus,
<em>v(t)</em> = <em>v(0)</em> + ∫₀ᵗ <em>a(u)</em> d<em>u</em>
The particle starts at rest, so <em>v(0)</em> = 0. Computing the integral gives
<em>v(t)</em> = [2/3 <em>u</em> ³ + 2<em>u</em> ²]₀ᵗ = 2/3 <em>t</em> ³ + 2<em>t</em> ²
(b) Use the FTC again, but this time you want the distance, which means you need to integrate the <u>speed</u> of the particle, i.e. the absolute value of <em>v(t)</em>. Fortunately, for <em>t</em> ≥ 0, we have <em>v(t)</em> ≥ 0 and |<em>v(t)</em> | = <em>v(t)</em>, so speed is governed by the same function. Taking the starting point to be the origin, after 8 seconds the particle travels a distance of
∫₀⁸ <em>v(u)</em> d<em>u</em> = ∫₀⁸ (2/3 <em>u</em> ³ + 2<em>u</em> ²) d<em>u</em> = [1/6 <em>u</em> ⁴ + 2/3 <em>u</em> ³]₀⁸ = 1024
