Question

1. determine whether the following functions are continuous, differentiable, both or neither at the point x = c. fill in the following table with yes or no based on your analysis.

a)
c)
e)
b)
d)
f)
function | continuous | differentiable | neither
a
b
c
d
e
f

1. find the derivative of the following function at x = a using the limit - definition of the derivative.

(a) f(x)=8 - 2x
(b) f(x)=2x²

Explanation:

Step1: Recall continuity and differentiability definitions

A function is continuous at a point if $\lim_{x
ightarrow c^{-}}f(x)=\lim_{x
ightarrow c^{+}}f(x)=f(c)$. A function is differentiable at a point if the left - hand derivative $\lim_{h
ightarrow0^{-}}\frac{f(c + h)-f(c)}{h}$ and the right - hand derivative $\lim_{h
ightarrow0^{+}}\frac{f(c + h)-f(c)}{h}$ are equal.

Step2: Analyze graphs for a - f

For graph a:

• Continuity: The graph has a break at $c$, so it is neither continuous nor differentiable.

For graph b:

• Continuity: The graph has a sharp corner at $c$, so it is continuous but not differentiable.

For graph c:

• Continuity: The graph is smooth and unbroken at $c$, so it is both continuous and differentiable.

For graph d:

• Continuity: The graph has a vertical asymptote at $c$, so it is neither continuous nor differentiable.

For graph e:

• Continuity: The graph has a jump at $c$, so it is neither continuous nor differentiable.

For graph f:

• Continuity: The graph is smooth and unbroken at $c$, so it is both continuous and differentiable.

Step3: Analyze derivative using limit definition for 3(a)

The limit definition of the derivative is $f^\prime(a)=\lim_{h
ightarrow0}\frac{f(a + h)-f(a)}{h}$. Given $f(x)=8 - 2x$, then $f(a)=8 - 2a$ and $f(a + h)=8-2(a + h)=8 - 2a-2h$.
\[

$$\begin{align*} f^\prime(a)&=\lim_{h ightarrow0}\frac{(8 - 2a-2h)-(8 - 2a)}{h}\\ &=\lim_{h ightarrow0}\frac{8 - 2a-2h - 8 + 2a}{h}\\ &=\lim_{h ightarrow0}\frac{-2h}{h}\\ &=- 2 \end{align*}$$

\]

Step4: Analyze derivative using limit definition for 3(b)

Given $f(x)=2x^{2}$, then $f(a)=2a^{2}$ and $f(a + h)=2(a + h)^{2}=2(a^{2}+2ah+h^{2})=2a^{2}+4ah + 2h^{2}$.
\[

$$\begin{align*} f^\prime(a)&=\lim_{h ightarrow0}\frac{2a^{2}+4ah + 2h^{2}-2a^{2}}{h}\\ &=\lim_{h ightarrow0}\frac{4ah+2h^{2}}{h}\\ &=\lim_{h ightarrow0}(4a + 2h)\\ &=4a \end{align*}$$

\]

Answer:

Function	Continuous	Differentiable	Neither
b	YES	NO	NO
c	YES	YES	NO
d	NO	NO	YES
e	NO	NO	YES
f	YES	YES	NO

3(a) Answer: $-2$
3(b) Answer: $4a$