Question

The table above gives selected values of twice-differentiable functions f and g, as well as the first two derivatives of g. If f′ (x)=3 for all values of x, what is the value of $\begin{equation}\int_{2}^{4} f(x) g^{\prime \prime}(x) d x\end{equation}$

Answer 1

Answer:

63

Step-by-step explanation:

∫₂⁴ f(x) g"(x) dx

Integrate by parts.

If u = f(x), then du = f'(x) dx.

If dv = g"(x) dx, then v = g'(x).

∫ u dv = uv − ∫ v du

= f(x) g'(x) − ∫ g'(x) f'(x) dx

f'(x) is constant at 3, so:

= f(x) g'(x) − ∫ 3 g'(x) dx

= f(x) g'(x) − 3 g(x)

Evaluate from x=2 to x=4.

[f(4) g'(4) − 3 g(4)] − [f(2) g'(2) − 3 g(2)]

[(13) (7) − 3 (9)] − [(7) (1) − 3 (2)]

64 − 1

63

Answer 2

Answer:

4009

Step-by-step explanation: