Question

assignment 10: problem 7 (1 point) consider the function ( f(x) ) whose second derivative is ( f(x) = 10x + 9sin(x) ). if ( f(0) = 4 ) and ( f(0) = 2 ), what is ( f(4) )? you have attempted this problem 0 times. you have unlimited attempts remaining.

Explanation:

Step1: Find \( f'(x) \) by integrating \( f''(x) \)

Given \( f''(x) = 10x + 9\sin(x) \), integrate with respect to \( x \):
\( f'(x)=\int (10x + 9\sin(x))dx = 5x^2 - 9\cos(x)+C \).
Use \( f'(0) = 2 \) to find \( C \):
\( f'(0)=5(0)^2 - 9\cos(0)+C = -9 + C = 2 \implies C = 11 \).
Thus, \( f'(x)=5x^2 - 9\cos(x)+11 \).

Step2: Find \( f(x) \) by integrating \( f'(x) \)

Integrate \( f'(x) \) with respect to \( x \):
\( f(x)=\int (5x^2 - 9\cos(x)+11)dx=\frac{5}{3}x^3 - 9\sin(x)+11x + D \).
Use \( f(0) = 4 \) to find \( D \):
\( f(0)=\frac{5}{3}(0)^3 - 9\sin(0)+11(0) + D = D = 4 \implies D = 4 \).
Thus, \( f(x)=\frac{5}{3}x^3 - 9\sin(x)+11x + 4 \).

Step3: Evaluate \( f(4) \)

Substitute \( x = 4 \) into \( f(x) \):
\( f(4)=\frac{5}{3}(4)^3 - 9\sin(4)+11(4) + 4 \)
\(=\frac{5}{3}(64) - 9\sin(4)+44 + 4 \)
\(=\frac{320}{3} - 9\sin(4)+48 \)
\(=\frac{320}{3}+\frac{144}{3} - 9\sin(4) \)
\(=\frac{464}{3} - 9\sin(4) \approx \frac{464}{3}-9(-0.7568) \) (since \( \sin(4)\approx -0.7568 \))
\(=\frac{464}{3}+6.8112 \approx 154.6667 + 6.8112 \approx 161.4779 \).

Answer:

\( \frac{464}{3} - 9\sin(4) \) (or approximately \( 161.48 \))