Question

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

Explanation:

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

Given \(F(x)=-x^{2}+35x + 291\), using the power rule \((x^{n})^\prime=nx^{n - 1}\), we have \(F^\prime(x)=-2x + 35\).

Step2: Calculate \(F^\prime(10)\) and \(F^\prime(15)\)

• For \(x = 10\), \(F^\prime(10)=-2\times10+35=15\).
• For \(x = 15\), \(F^\prime(15)=-2\times15+35 = 5\).

Step3: Recognize the shape of \(y = F^\prime(x)\)

Since \(F^\prime(x)=-2x + 35\) is a linear function (\(y=mx + b\) form where \(m=-2\) and \(b = 35\)), its graph is a straight - line.

Step4: Calculate the area between the graph of \(y = F^\prime(x)\) and the \(x\) - axis from \(x = 10\) to \(x = 15\)

The area \(A\) between the line \(y=-2x + 35\), \(x = 10\), \(x = 15\) and the \(x\) - axis is the area of a trapezoid. The formula for the area of a trapezoid is \(A=\frac{(a + b)h}{2}\), where \(a=F^\prime(10)=15\), \(b=F^\prime(15)=5\) and \(h=15 - 10=5\).

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Answer:

The area is \(50\).