use the four - step process to find f(x) and then find f(1), f(2), and f(3).\nf(x)=x^{2}+4x - 5\nf(x)= \n…

use the four - step process to find f(x) and then find f(1), f(2), and f(3).\nf(x)=x^{2}+4x - 5\nf(x)= \n f(1)= (type an integer or a simplified fraction.)\n f(2)= (type an integer or a simplified fraction.)\n f(3)= (type an integer or a simplified fraction.)

use the four - step process to find f(x) and then find f(1), f(2), and f(3).\nf(x)=x^{2}+4x - 5\nf(x)= \n f(1)= (type an integer or a simplified fraction.)\n f(2)= (type an integer or a simplified fraction.)\n f(3)= (type an integer or a simplified fraction.)

Answer

Explanation:

Step1: Find $f(x + h)$

$f(x+h)=(x + h)^2+4(x + h)-5=x^{2}+2xh+h^{2}+4x + 4h-5$

Step2: Find $f(x + h)-f(x)$

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

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

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

Step4: Find $\lim_{h\rightarrow0}\frac{f(x + h)-f(x)}{h}$

[ \begin{align*} f^{\prime}(x)&=\lim_{h\rightarrow0}(2x+h + 4)\ &=2x+4 \end{align*} ] To find $f^{\prime}(1)$: Substitute $x = 1$ into $f^{\prime}(x)$: $f^{\prime}(1)=2\times1+4=6$ To find $f^{\prime}(2)$: Substitute $x = 2$ into $f^{\prime}(x)$: $f^{\prime}(2)=2\times2+4=8$ To find $f^{\prime}(3)$: Substitute $x = 3$ into $f^{\prime}(x)$: $f^{\prime}(3)=2\times3+4=10$

Answer:

$f^{\prime}(x)=2x + 4$ $f^{\prime}(1)=6$ $f^{\prime}(2)=8$ $f^{\prime}(3)=10$