hw 7 - definition of the derivative section 2.2: problem 4 (1 point)\nlet $f(x)=5x - 2x^{2}$. if $h\neq0$…

hw 7 - definition of the derivative section 2.2: problem 4 (1 point)\nlet $f(x)=5x - 2x^{2}$. if $h\neq0$, then the difference quotient can be simplified as\n$\frac{f(x + h)-f(x)}{h}=ah + bx + c$,\nwhere $a$, $b$, and $c$ are constants. (note: its possible for one or more of these constants to be 0.) find the constants.\n$a=square$, $b=square$, and $c=square$.\nuse the simplified expression to find $f(x)=lim_{h\rightarrow0}\frac{f(x + h)-f(x)}{h}=square$.\nfinally, find each of the following.\n$f(1)=square$, $f(2)=square$, and $f(3)=square$.

hw 7 - definition of the derivative section 2.2: problem 4 (1 point)\nlet $f(x)=5x - 2x^{2}$. if $h\neq0$, then the difference quotient can be simplified as\n$\frac{f(x + h)-f(x)}{h}=ah + bx + c$,\nwhere $a$, $b$, and $c$ are constants. (note: its possible for one or more of these constants to be 0.) find the constants.\n$a=square$, $b=square$, and $c=square$.\nuse the simplified expression to find $f(x)=lim_{h\rightarrow0}\frac{f(x + h)-f(x)}{h}=square$.\nfinally, find each of the following.\n$f(1)=square$, $f(2)=square$, and $f(3)=square$.

Answer

Explanation:

Step1: Find (f(x + h))

[ \begin{align*} f(x+h)&=5(x + h)-2(x + h)^{2}\ &=5x+5h-2(x^{2}+2xh + h^{2})\ &=5x+5h-2x^{2}-4xh - 2h^{2} \end{align*} ]

Step2: Calculate (\frac{f(x + h)-f(x)}{h})

[ \begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{(5x+5h-2x^{2}-4xh - 2h^{2})-(5x - 2x^{2})}{h}\ &=\frac{5x+5h-2x^{2}-4xh - 2h^{2}-5x + 2x^{2}}{h}\ &=\frac{5h-4xh - 2h^{2}}{h}\ &=- 2h-4x + 5 \end{align*} ] Comparing with (Ah + Bx + C), we have (A=-2), (B=-4), (C = 5)

Step3: Find (f'(x))

[ \begin{align*} f'(x)&=\lim_{h\rightarrow0}\frac{f(x + h)-f(x)}{h}\ &=\lim_{h\rightarrow0}(-2h-4x + 5)\ &=-4x + 5 \end{align*} ]

Step4: Find (f'(1)), (f'(2)) and (f'(3))

[ \begin{align*} f'(1)&=-4\times1 + 5=1\ f'(2)&=-4\times2+5=-3\ f'(3)&=-4\times3 + 5=-7 \end{align*} ]

Answer:

(A=-2), (B=-4), (C = 5), (f'(x)=-4x + 5), (f'(1)=1), (f'(2)=-3), (f'(3)=-7)