If $f(x)=a x^2+b x+c$ and $f\left(\frac{1}{n}\right)=\frac{n+1}{n^2}$ for all $n \in N$, then what i...

Given the function:$ f(x) = ax^2 + bx + c $and that:$ f\left(\frac{1}{n}\right) = \frac{n+1}{n^2} $for all $ n \in \mathbb{N} $.We need to determine the value of:$ \lim_{x \rightarrow 0} f^{\prime}(x) $First, compute the derivative of $ f(x) $:$ f'(x) = 2ax + b $Thus:$ f'(0) = b $To find $ b $, match $ f\left( \frac{1}{n} \right) $ with:$ \frac{1+n}{n^2} = \frac{1}{n^2} + \frac{1}{n} $Introduce the substitution $ \frac{1}{n} = t $, so:$ f(t) = t^2 + t + 0 $This shows that:$ b = 1 $The derivative at 0 is thus:$ f'(0) = b = 1 $Therefore, the value of $ \lim_{x \rightarrow 0} f^\prime(x) $ is 1.

Which one of the following functions is differentiable at $x=0$ ?...

If a curve is smooth at a point, it has no sharp turns or discontinuities, meaning a consistent tangent can be drawn. This ensures continuity and a unique slope at the point, fulfilling the conditions for differentiability. Thus, smoothness at a point guarantees the curve is differentiable there.Shortcut/advice for imagining graph: Here, The $|x|$ makes the graph mirror itself about the $y$-axis(. As $\cos (x)$ is the same around $y$-axis, it behaves the same as $\cos |x|$.

Let $f, g: R \rightarrow R$ be functions. If $g$ is continuous, then which one of the following case...

When considering whether a function $ f $ is continuous based on the continuity of another function $ g $, where $ g $ is expressed in terms of $ f $, we evaluate several scenarios:Case: $ g(x) = (f(x))^3 $For $ g(x) = (f(x))^3 $ to be continuous, $ f(x) $ must also be continuous. This is because the composition of continuous functions (in this case, the cubic function and $ f(x) $) results in a continuous function.Case: $ g(x) = |f(x)| $Consider the function:$ f(x) = \begin{cases} 2 & \text{if } x \geq 0 \\ -2 & \text{if } x < 0 \end{cases} $Here, $|f(x)|$ is continuous and equals 2 for any value of $ x $, even though $ f(x) $ itself is not continuous due to the jump discontinuity at $ x = 0 $.Case: $ g(x) = (f(x))^2 $Using the same function $ f(x) $ as in the previous case:$ f(x) = \begin{cases} 2 & \text{if } x \geq 0 \\ -2 & \text{if } x < 0 \end{cases} $Here, $(f(x))^2$ is continuous and equals 4 for any value of $ x $, while $ f(x) $ is not continuous.Case: $ g(x) = \sin(f(x)) $Consider the function:$ f(x) = \begin{cases} \frac{\pi}{2} & \text{if } x \geq 0 \\ \frac{5\pi}{4} & \text{if } x < 0 \end{cases} $In this case, $\sin(f(x))$ is continuous and equals 1 for any value of $ x $, whereas $ f(x) $ is not continuous due to the jump at $ x = 0 $.In summary, while $ g(x) $ is continuous in the forms evaluated, only in the case of $ g(x) = (f(x))^3 $ does this directly imply $ f(x) $ itself is continuous. The other forms allow $ g(x) $ to be continuous even if $ f(x) $ is not.

Limits, Continuity And Differentiability

Given the function:

f(x)%20%3D%20ax%5E2%20%2B%20bx%20%2B%20c

and that:

f%5Cleft(%5Cfrac%7B1%7D%7Bn%7D%5Cright)%20%3D%20%5Cfrac%7Bn%2B1%7D%7Bn%5E2%7D

for all

n%20%5Cin%20%5Cmathbb%7BN%7D

.We need to determine the value of:

%5Clim_%7Bx%20%5Crightarrow%200%7D%20f%5E%7B%5Cprime%7D(x)

First, compute the derivative of

f(x)

f'(x)%20%3D%202ax%20%2B%20b

Thus:

f'(0)%20%3D%20b

To find

b

, match

f%5Cleft(%20%5Cfrac%7B1%7D%7Bn%7D%20%5Cright)

with:

%5Cfrac%7B1%2Bn%7D%7Bn%5E2%7D%20%3D%20%5Cfrac%7B1%7D%7Bn%5E2%7D%20%2B%20%5Cfrac%7B1%7D%7Bn%7D

Introduce the substitution

%5Cfrac%7B1%7D%7Bn%7D%20%3D%20t

, so:

f(t)%20%3D%20t%5E2%20%2B%20t%20%2B%200

This shows that:

b%20%3D%201

The derivative at 0 is thus:

f'(0)%20%3D%20b%20%3D%201

Therefore, the value of

%5Clim_%7Bx%20%5Crightarrow%200%7D%20f%5E%5Cprime(x)

is 1.

When considering whether a function

f

is continuous based on the continuity of another function

g

, where

g

is expressed in terms of

f

, we evaluate several scenarios:Case:

g(x)%20%3D%20(f(x))%5E3

For

g(x)%20%3D%20(f(x))%5E3

to be continuous,

f(x)

must also be continuous. This is because the composition of continuous functions (in this case, the cubic function and

f(x)

) results in a continuous function.Case:

g(x)%20%3D%20%7Cf(x)%7C

Consider the function:

f(x)%20%3D%20%5Cbegin%7Bcases%7D%202%20%26%20%5Ctext%7Bif%20%7D%20x%20%5Cgeq%200%20%5C%5C%20-2%20%26%20%5Ctext%7Bif%20%7D%20x%20%3C%200%20%5Cend%7Bcases%7D

Here,

%7Cf(x)%7C

is continuous and equals 2 for any value of

x

, even though

f(x)

itself is not continuous due to the jump discontinuity at

x%20%3D%200

.Case:

g(x)%20%3D%20(f(x))%5E2

Using the same function

f(x)

as in the previous case:

f(x)%20%3D%20%5Cbegin%7Bcases%7D%202%20%26%20%5Ctext%7Bif%20%7D%20x%20%5Cgeq%200%20%5C%5C%20-2%20%26%20%5Ctext%7Bif%20%7D%20x%20%3C%200%20%5Cend%7Bcases%7D

Here,

(f(x))%5E2

is continuous and equals 4 for any value of

x

, while

f(x)

is not continuous.Case:

g(x)%20%3D%20%5Csin(f(x))

Consider the function:

f(x)%20%3D%20%5Cbegin%7Bcases%7D%20%5Cfrac%7B%5Cpi%7D%7B2%7D%20%26%20%5Ctext%7Bif%20%7D%20x%20%5Cgeq%200%20%5C%5C%20%5Cfrac%7B5%5Cpi%7D%7B4%7D%20%26%20%5Ctext%7Bif%20%7D%20x%20%3C%200%20%5Cend%7Bcases%7D

In this case,

%5Csin(f(x))

is continuous and equals 1 for any value of

x

, whereas

f(x)

is not continuous due to the jump at

x%20%3D%200

.In summary, while

g(x)

is continuous in the forms evaluated, only in the case of

g(x)%20%3D%20(f(x))%5E3

does this directly imply

f(x)

itself is continuous. The other forms allow

g(x)

to be continuous even if

f(x)

is not.

Limits, Continuity And Differentiability

High-Yield Trend

Questions
4 MCQs

Official Solution

Official Solution

Official Solution

Official Solution

Limits, Continuity And Differentiability

High-Yield Trend

Questions 4 MCQs

Official Solution

Official Solution

Official Solution

Official Solution

Questions
4 MCQs