Question

answer the following questions about ( f(x)=-x^{2}+35x + 291 ).
the change is 50.
(b) graph ( f^{prime}(x) ) and use geometric formulas to calculate the area between the graph of ( f^{prime}(x) ) and the ( x )-axis from ( x = 10 ) and ( x = 15 ).
choose the correct graph below.
the area between the graph of ( f^{prime}(x) ) and the ( x )-axis from ( x = 10 ) to ( x = 15 ) is

Explanation:

Step1: Find the derivative of \(F(x)\)

Using the power rule \((x^n)^\prime=nx^{n - 1}\), if \(F(x)=-x^{2}+35x + 291\), then \(F^\prime(x)=-2x + 35\).
When \(x = 10\), \(F^\prime(10)=-2\times10 + 35=15\).
When \(x = 15\), \(F^\prime(15)=-2\times15+35 = 5\).

Step2: Recognize the geometric shape

The function \(y = F^\prime(x)=-2x + 35\) is a linear function. The region between \(x = 10\) and \(x = 15\) under the line \(y=-2x + 35\) and above the \(x\)-axis is a trapezoid.
The formula for the area of a trapezoid is \(A=\frac{(a + b)h}{2}\), where \(a\) and \(b\) are the lengths of the parallel sides and \(h\) is the height (the difference in the \(x\)-values). Here, \(a = F^\prime(10)=15\), \(b = F^\prime(15)=5\), and \(h=15 - 10=5\).

Step3: Calculate the area

Substitute the values into the trapezoid - area formula:
\(A=\frac{(15 + 5)\times5}{2}=\frac{20\times5}{2}=50\).

Answer:

\(50\)