The curved section can be modelled by the parabola
y=3x-3/4x^2
Use calculus to find the maximum height of the curved parabolic section.
How do I do this step by step?
If you have done functions before calculus as most students do, you can rewrite the equation in the standard form:
f(x)=a(x-h)+k
where the vertex is found at
(h,k)
or
f(x)=-(3/4)(x-2)²+3
where the vertex is found at (2,3).
Confirm by Calculus:
f(x)=3x-3/4x^2
f'(x)=3-(3/2)x
At the vertex, f'(x)=0
x=3*(2/3)=2
y=3(2)-(3/4)(2²)
=6-3
=3
So the vertex is at (2,3)
your equation is ambiguous
do you mean
y = (3x-3)/(4x^2) ? or
y = 3x - 3/(4x^2) ? or
y = ((3x-3)/4)(x^2) or
y = 3x - (3/4)x^2 ? or ....
go with Damon
I skimmed over the "parabolic section" of your post.
..The last one.
Confirm by Calculus:
f(x)=3x-3/4x^2
f'(x)=3-(3/2)x
At the vertex, f'(x)=0
x=3*(2/3)=2
y=3(2)-(3/4)(2²)
=6-3
=3
So the vertex is at (2,3)
A question regarding the first step (differentiation) How do you differentiate the fractions?
I'm stuck.. And clearly thick. Haha.
You do not really differentiate the fractional numbers, they are just the coefficients which just "stick around".
f(x)=ax²+bx
f'(x)=2ax+b
So in the above,
a=-(3/4), b=3
f'(x)=2(-3/4)x+3
=-(3/2)x+3
Okay, thank you very much!
You're welcome!
I had a run over it myself and got the maxima as (-2,-3) same numbers, but negative form?
Check your work, draw a graphical sketch to convince yourself that there is only one maximum.
Also, f(2)=3, and f(-2)=-9, so (-2,-3) does not lie on the curve.
If all fails, post your working for a check.
To find the maximum height of the curved parabolic section, you need to find the vertex of the parabola. The vertex represents the highest point on the curve.
Step 1: Write the equation of the parabola in standard form.
Rewrite the equation of the parabola y = 3x - (3/4)x^2 in standard form. This can be done by completing the square or by factoring.
Step 2: Determine the coefficients of the quadratic equation.
Identify the coefficients a, b, and c in the general form of a quadratic equation ax^2 + bx + c. In this case, a = -3/4, b = 3, and c = 0.
Step 3: Find the x-coordinate of the vertex.
The x-coordinate of the vertex can be determined using the formula: x = -b / (2a). Substituting the values of a and b, we get: x = -3 / (2 * -3/4). Simplifying this equation, we find x = 2.
Step 4: Find the y-coordinate of the vertex.
Plug the value of x (found in Step 3) into the equation y = 3x - (3/4)x^2 to determine the y-coordinate of the vertex. Substituting x = 2 into the equation, we get: y = 3(2) - (3/4)(2^2) = 6 - (3/4)(4) = 6 - 3 = 3.
Step 5: Determine the maximum height.
The vertex (2, 3) represents the maximum height of the curved parabolic section. Therefore, the maximum height is 3 units.
By following these steps and applying calculus concepts, you can determine the maximum height of the curved parabolic section modeled by the given equation.