Question

if ( f(6)=10 ), ( f^{prime} ) is continuous, and ( int_{6}^{7} f^{prime}(x) d x=15 ), what is the value of ( f(7) )?
( f(7)= )

Explanation:

Step1: Use the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus states that if \(F(x)\) is an antiderivative of \(f(x)\), then \(\int_{a}^{b}f(x)dx = F(b)-F(a)\). Here, since \(f(x)\) is an antiderivative of \(f^{\prime}(x)\), we have \(\int_{6}^{7}f^{\prime}(x)dx=f(7)-f(6)\).

Step2: Substitute the given values

We know that \(f(6) = 10\) and \(\int_{6}^{7}f^{\prime}(x)dx = 15\). Substituting these into the equation \(f(7)-f(6)=\int_{6}^{7}f^{\prime}(x)dx\), we get \(f(7)-10 = 15\).

Step3: Solve for \(f(7)\)

Add \(10\) to both sides of the equation \(f(7)-10 = 15\). So, \(f(7)=15 + 10\).

Answer:

\(25\)