The park will have 600 gazelles, you just gotta miltiply 120 and 5 :)
<h2>
Part 1.</h2>
Answer: ![\boxed{f'(x)=3x^2-8x}](https://tex.z-dn.net/?f=%5Cboxed%7Bf%27%28x%29%3D3x%5E2-8x%7D)
The limit of the difference quotient is simply the derivative, so we can express this as follows:
![f'(x)=\underset{\Delta x\rightarrow0}{lim}\frac{\triangle y}{\Delta x}=\frac{f(x+\Delta x)-f(x)}{\Delta x}](https://tex.z-dn.net/?f=f%27%28x%29%3D%5Cunderset%7B%5CDelta%20x%5Crightarrow0%7D%7Blim%7D%5Cfrac%7B%5Ctriangle%20y%7D%7B%5CDelta%20x%7D%3D%5Cfrac%7Bf%28x%2B%5CDelta%20x%29-f%28x%29%7D%7B%5CDelta%20x%7D)
So our function is:
![f(x)=x^3-4x^2+2](https://tex.z-dn.net/?f=f%28x%29%3Dx%5E3-4x%5E2%2B2)
Taking the derivative, we have:
![f'(x)=3x^2-8x](https://tex.z-dn.net/?f=f%27%28x%29%3D3x%5E2-8x)
So the correct option is:
![\boxed{f'(x)=3x^2-8x}](https://tex.z-dn.net/?f=%5Cboxed%7Bf%27%28x%29%3D3x%5E2-8x%7D)
<h2>Part 2.</h2>
Answer: ![\boxed{y=3x-16}](https://tex.z-dn.net/?f=%5Cboxed%7By%3D3x-16%7D)
The equation of the line that passes through the same point can be found as:
![y-y_{0}=m(x-x_{0})](https://tex.z-dn.net/?f=y-y_%7B0%7D%3Dm%28x-x_%7B0%7D%29)
Where
, so we need to find
. Plug in that x-value in the function we have:
![y_{0}=f(3)=(3)^3-4(3)^2+2 \\ \\ y_{0}=-7 \\ \\ \\ So \ the \ point \ is: \\ \\ P(x_{0},y_{0})=(3,-7)](https://tex.z-dn.net/?f=y_%7B0%7D%3Df%283%29%3D%283%29%5E3-4%283%29%5E2%2B2%20%5C%5C%20%5C%5C%20y_%7B0%7D%3D-7%20%5C%5C%20%5C%5C%20%5C%5C%20So%20%5C%20the%20%5C%20point%20%5C%20is%3A%20%5C%5C%20%5C%5C%20P%28x_%7B0%7D%2Cy_%7B0%7D%29%3D%283%2C-7%29)
And the slope is:
![m=f'(3)=3(3)^2-8(3) \\ \\ m=3](https://tex.z-dn.net/?f=m%3Df%27%283%29%3D3%283%29%5E2-8%283%29%20%5C%5C%20%5C%5C%20m%3D3)
Then, the equation of the line is:
![y-(-7)=3(x-3) \\ \\ \therefore y+7=3x-9 \\ \\ \boxed{y=3x-16}](https://tex.z-dn.net/?f=y-%28-7%29%3D3%28x-3%29%20%5C%5C%20%5C%5C%20%5Ctherefore%20y%2B7%3D3x-9%20%5C%5C%20%5C%5C%20%5Cboxed%7By%3D3x-16%7D)
<h2 /><h2>Part 3.</h2>
Answer: Shown below
As you can see below, the graph of the function of
is continuous. This is so because we have plotted a polynomial function whose domain is the set of all real numbers. So the function is defined at the point
, so the derivative exists at this point, hence we can calculate a tangent line there. In conclusion, we get the graph shown below. The blue line is the tangent line while the red curve is the graph of ![f](https://tex.z-dn.net/?f=f)
49 and I really can't say how I got it cuz it's too much work
Answer:
I think it's a because when if the graph was actually able to pass the y axis before the x axis it would be ( 10, 0) however since it's the x axis coming first and it is negative it is going to be ( 0 , -10 )