select the correct expressions. identify each expression that represents the slope of a tangent to the curve…

select the correct expressions. identify each expression that represents the slope of a tangent to the curve y = -x³ + 17x² - x + 3 at any point (x, y). lim h→0 (-3x²h - 3xh² - h³ + 34xh + 17h² - 5h)/h -3x² + 34x - 1 lim h→0 -x² - 6xh - h² + 17x + 17h - 1 -3x² + 34x - 5 lim h→0 (-3x²h - 3xh² - h³ + 34xh + 17h² - h)/h lim h→0 -3x² - 3xh - h² + 34x + 17h - 1 -x² + 17x - 1 lim h→0 (-x²h - 6xh² - h³ + 17xh + 17h² - h)/h

select the correct expressions. identify each expression that represents the slope of a tangent to the curve y = -x³ + 17x² - x + 3 at any point (x, y). lim h→0 (-3x²h - 3xh² - h³ + 34xh + 17h² - 5h)/h -3x² + 34x - 1 lim h→0 -x² - 6xh - h² + 17x + 17h - 1 -3x² + 34x - 5 lim h→0 (-3x²h - 3xh² - h³ + 34xh + 17h² - h)/h lim h→0 -3x² - 3xh - h² + 34x + 17h - 1 -x² + 17x - 1 lim h→0 (-x²h - 6xh² - h³ + 17xh + 17h² - h)/h

Answer

Explanation:

Step1: Recall the derivative formula

The slope of the tangent to the curve $y = f(x)$ at a point $(x,y)$ is given by $f^\prime(x)=\lim_{h\rightarrow0}\frac{f(x + h)-f(x)}{h}$. Given $y=f(x)=-x^{3}+17x^{2}-x + 3$, then $f(x + h)=-(x + h)^{3}+17(x + h)^{2}-(x + h)+3$.

Step2: Expand $f(x + h)$

[ \begin{align*} f(x + h)&=-(x^{3}+3x^{2}h + 3xh^{2}+h^{3})+17(x^{2}+2xh+h^{2})-x - h+3\ &=-x^{3}-3x^{2}h-3xh^{2}-h^{3}+17x^{2}+34xh + 17h^{2}-x - h+3 \end{align*} ]

Step3: Calculate $f(x + h)-f(x)$

[ \begin{align*} f(x + h)-f(x)&=(-x^{3}-3x^{2}h-3xh^{2}-h^{3}+17x^{2}+34xh + 17h^{2}-x - h+3)-(-x^{3}+17x^{2}-x + 3)\ &=-3x^{2}h-3xh^{2}-h^{3}+34xh + 17h^{2}-h \end{align*} ]

Step4: Calculate $\frac{f(x + h)-f(x)}{h}$

[ \frac{f(x + h)-f(x)}{h}=\frac{-3x^{2}h-3xh^{2}-h^{3}+34xh + 17h^{2}-h}{h}=-3x^{2}-3xh - h^{2}+34x + 17h-1 ]

Step5: Find the limit as $h\rightarrow0$

[ \lim_{h\rightarrow0}\frac{f(x + h)-f(x)}{h}=\lim_{h\rightarrow0}(-3x^{2}-3xh - h^{2}+34x + 17h-1)=-3x^{2}+34x - 1 ] Also, $\lim_{h\rightarrow0}\frac{-3x^{2}h-3xh^{2}-h^{3}+34xh + 17h^{2}-h}{h}$ is the correct limit - based expression for the derivative. And $\lim_{h\rightarrow0}-3x^{2}-3xh - h^{2}+34x + 17h-1$ is also correct as it is the simplified form of the limit before taking $h\rightarrow0$.

Answer:

$\lim_{h\rightarrow0}\frac{-3x^{2}h-3xh^{2}-h^{3}+34xh + 17h^{2}-h}{h}$, $-3x^{2}+34x - 1$, $\lim_{h\rightarrow0}-3x^{2}-3xh - h^{2}+34x + 17h-1$