Question

1. -/1 points if (f(x)=\frac{x^{2}}{2 + x}), find (f(2)). (f(2)=) blank resources read it watch it submit answer 14. -/3 points suppose that (f(5)=1,f(5)=5,g(5)= - 2), and (g(5)=3). find the following values. (a) ((fg)(5)) blank (b) ((\frac{f}{g})(5)) blank (c) ((\frac{g}{f})(5)) blank

Explanation:

Step1: Find the first - derivative of \(f(x)=\frac{x^{2}}{2 + x}\) using the quotient rule

The quotient rule states that if \(y=\frac{u}{v}\), then \(y'=\frac{u'v - uv'}{v^{2}}\). Here, \(u = x^{2}\), \(u'=2x\), \(v=2 + x\), \(v' = 1\). So \(f'(x)=\frac{2x(2 + x)-x^{2}(1)}{(2 + x)^{2}}=\frac{4x+2x^{2}-x^{2}}{(2 + x)^{2}}=\frac{x^{2}+4x}{(2 + x)^{2}}\).

Step2: Find the second - derivative of \(f(x)\) using the quotient rule again

Let \(u=x^{2}+4x\), \(u'=2x + 4\), \(v=(2 + x)^{2}\), \(v'=2(2 + x)\). Then \(f''(x)=\frac{(2x + 4)(2 + x)^{2}-(x^{2}+4x)\times2(2 + x)}{(2 + x)^{4}}=\frac{(2x + 4)(2 + x)-2(x^{2}+4x)}{(2 + x)^{3}}=\frac{4x+2x^{2}+8 + 4x-2x^{2}-8x}{(2 + x)^{3}}=\frac{8}{(2 + x)^{3}}\).

Step3: Evaluate \(f''(2)\)

Substitute \(x = 2\) into \(f''(x)\): \(f''(2)=\frac{8}{(2 + 2)^{3}}=\frac{8}{64}=\frac{1}{8}\).

for 14(a):

Step1: Use the product rule \((fg)'(x)=f'(x)g(x)+f(x)g'(x)\)

We know that \(f(5) = 1\), \(f'(5)=5\), \(g(5)=-2\), and \(g'(5)=3\). Substitute these values into the product - rule formula: \((fg)'(5)=f'(5)g(5)+f(5)g'(5)\).

Step2: Calculate the value

\((fg)'(5)=5\times(-2)+1\times3=-10 + 3=-7\).

for 14(b):

Step1: Use the quotient rule \((\frac{f}{g})'(x)=\frac{f'(x)g(x)-f(x)g'(x)}{g^{2}(x)}\)

Substitute \(x = 5\), \(f(5) = 1\), \(f'(5)=5\), \(g(5)=-2\), and \(g'(5)=3\) into the quotient - rule formula: \((\frac{f}{g})'(5)=\frac{f'(5)g(5)-f(5)g'(5)}{g^{2}(5)}\).

Step2: Calculate the value

\((\frac{f}{g})'(5)=\frac{5\times(-2)-1\times3}{(-2)^{2}}=\frac{-10 - 3}{4}=-\frac{13}{4}\).

for 14(c):

Step1: Use the quotient rule \((\frac{g}{f})'(x)=\frac{g'(x)f(x)-g(x)f'(x)}{f^{2}(x)}\)

Substitute \(x = 5\), \(f(5) = 1\), \(f'(5)=5\), \(g(5)=-2\), and \(g'(5)=3\) into the quotient - rule formula: \((\frac{g}{f})'(5)=\frac{g'(5)f(5)-g(5)f'(5)}{f^{2}(5)}\).

Step2: Calculate the value

\((\frac{g}{f})'(5)=\frac{3\times1-(-2)\times5}{1^{2}}=\frac{3 + 10}{1}=13\).

Answer:

\(\frac{1}{8}\)