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

hw 7 - definition of the derivative section 2.2: problem 7 (1 point)\nlet $f(x)=-4 - 5x + 4x^{2}$. 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 $f(x)=\\lim_{h\\to0}\\frac{f(x + h)-f(x)}{h}=\\square$\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 7 (1 point)\nlet $f(x)=-4 - 5x + 4x^{2}$. 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 $f(x)=\\lim_{h\\to0}\\frac{f(x + h)-f(x)}{h}=\\square$\nfinally, find each of the following.\n$f(1)=\\square$, $f(2)=\\square$, and $f(3)=\\square$

Answer

Explanation:

Step1: Find (f(x + h))

Given (f(x)=-4 - 5x+4x^{2}), then (f(x + h)=-4-5(x + h)+4(x + h)^{2}). Expand (4(x + h)^{2}=4(x^{2}+2xh+h^{2}) = 4x^{2}+8xh + 4h^{2}). So (f(x + h)=-4-5x-5h + 4x^{2}+8xh+4h^{2}).

Step2: Calculate (f(x + h)-f(x))

[ \begin{align*} f(x + h)-f(x)&=(-4-5x-5h + 4x^{2}+8xh+4h^{2})-(-4 - 5x+4x^{2})\ &=-4-5x-5h + 4x^{2}+8xh+4h^{2}+4 + 5x-4x^{2}\ &=-5h+8xh + 4h^{2} \end{align*} ]

Step3: Find (\frac{f(x + h)-f(x)}{h})

[ \begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{-5h+8xh + 4h^{2}}{h}\ &=-5 + 8x+4h \end{align*} ] Comparing with (Ah + Bx+C), we have (A = 4), (B = 8), (C=-5).

Step4: Find (f'(x))

[ \begin{align*} f'(x)&=\lim_{h\rightarrow0}\frac{f(x + h)-f(x)}{h}\ &=\lim_{h\rightarrow0}(4h + 8x-5)\ &=8x-5 \end{align*} ]

Step5: Find (f'(1)), (f'(2)) and (f'(3))

When (x = 1), (f'(1)=8\times1-5=3). When (x = 2), (f'(2)=8\times2-5 = 11). When (x = 3), (f'(3)=8\times3-5=19).

Answer:

(A = 4), (B = 8), (C=-5), (f'(x)=8x - 5), (f'(1)=3), (f'(2)=11), (f'(3)=19)