Informative line

### Exponential Function And Value Of E

Practice Limits of Exponential & Graphs of Exponential Function & e Calculus, finding the inverse of a function which contain exponential function in them.

# Exponential Function

• If $$a>0$$ and $$a\neq1$$ then $$f(x)=a^x$$ is a continuous function whose domain is $$k$$ and range is $$(0,\,\infty)$$.
• $$a^x>0$$ for all real values of $$x$$.
• $$a^x$$ is an increasing function, if $$a>1$$
• $$a^x$$ is an decreasing function, if $$0<a<1$$

• ?$$a^n=a×a×a×.......n$$ times  if $$n\,\varepsilon N$$
• $$a^0=1$$ for all $$a$$.
• $$a^{-n}=\dfrac{1}{a^n}$$  $$n\,\varepsilon N$$
• $$a^{p/q}=\left(a^{1/q}\right)^p$$?

#### Find the exponential function of the from $$f(x)=k\,a^x$$, whose graph is as shown.

A $$f(x)=2\left(3^x\right)$$

B $$f(x)=4\left(5^x\right)$$

C $$f(x)=3\left(4^x\right)$$

D $$f(x)=2\left(5^x\right)$$

×

f(x) is an increasing function graph $$\Rightarrow \,a>1.$$

From the graph

$$\to f(0)=3$$

$$\Rightarrow\,3=k×a^0$$

$$\Rightarrow\,k=3$$

From the graph,

$$\to$$ $$f(2)=48$$

$$\Rightarrow\,48=k×a^2$$

$$\Rightarrow\,48=3a^2$$

$$\Rightarrow\,a^2=16$$

$$\Rightarrow\,a=4$$ (reject –4)

$$\therefore\,f(x)=3×4^x$$

### Find the exponential function of the from $$f(x)=k\,a^x$$, whose graph is as shown.

A

$$f(x)=2\left(3^x\right)$$

.

B

$$f(x)=4\left(5^x\right)$$

C

$$f(x)=3\left(4^x\right)$$

D

$$f(x)=2\left(5^x\right)$$

Option C is Correct

# Graphs of f(x)=ax

$$(0,\,1)$$ is a point on all the graphs of form $$f(x)=a^x$$ $$(0<a<1)$$

• Graph of $$f(x)=a^x$$ is a decreasing function of $$(0,\,1)$$.
• Graph of $$f$$ decreases more rapidly as $$'a'$$ increases.

• All the graphs of function of the form

$$f(x)=a^x$$ pass through $$(0,\,1)$$ as $$a°=1$$  $$\forall\;'a'$$.

• The graph grows more rapidly as value of a increases.

This is due to the fact that the number greater than 1 when raised to power will keep on increasing while numbers between 0 and 1 when raised to powers will keep on decreasing.

e.g. $$(0.2)^2=0.04$$ and $$(0.2)^3=0.008$$

while  $$(1.2)^2=1.44\;\;$$and  $$(1.2)^3=1.728$$

• $$\therefore\,a^x$$ is an increasing function of $$x$$,  when $$a>1$$ and it is a decreasing function of $$x$$, when $$0<a<1$$

#### Consider the graph of there functions on the same $$x-y$$ axis. Which of the following is the correct statement ?

A (1)  is the graph of $$f(x)=(1.7)^x$$ (2)  is the graph of $$f(x)=3^x$$ (3)  is the graph of $$f(x)=5^x$$

B (1)  is the graph of $$f(x)=7^x$$ (2)  is the graph of $$f(x)=3^x$$ (3)  is the graph of $$f(x)=(1.2)^x$$

C (1)  is the graph of $$f(x)=5^x$$ (2)  is the graph of $$f(x)=2 ^x$$ (3)  is the graph of $$f(x)=10^x$$

D (1)  is the graph of $$f(x)=(1.8)^x$$ (2)  is the graph of $$f(x)=10^x$$ (3)  is the graph of $$f(x)=5^x$$

×

(3)  is the steepest, then (2) and (1) is the slowest growing graph.

$$\therefore$$ 'a' should be greatest for (3) and least for (1)

$$\therefore$$ Hence, option (A) is correct.

### Consider the graph of there functions on the same $$x-y$$ axis. Which of the following is the correct statement ?

A

(1)  is the graph of $$f(x)=(1.7)^x$$

(2)  is the graph of $$f(x)=3^x$$

(3)  is the graph of $$f(x)=5^x$$

.

B

(1)  is the graph of $$f(x)=7^x$$

(2)  is the graph of $$f(x)=3^x$$

(3)  is the graph of $$f(x)=(1.2)^x$$

C

(1)  is the graph of $$f(x)=5^x$$

(2)  is the graph of $$f(x)=2 ^x$$

(3)  is the graph of $$f(x)=10^x$$

D

(1)  is the graph of $$f(x)=(1.8)^x$$

(2)  is the graph of $$f(x)=10^x$$

(3)  is the graph of $$f(x)=5^x$$

Option A is Correct

# Definition of the Number $$e$$

• $$e$$ is a number such that

$$\lim\limits_{h\to 0}\left(\dfrac{e^h-1}{h}\right)=1$$

• Consider,

$$\dfrac{d}{dx}(a^x)=\lim\limits_{h\to 0}\left(\dfrac{a^{x+h}-a^x}{h}\right)$$

$$=a^x\,\lim\limits_{h\to 0}\left(\dfrac{a^h-1}{h}\right)$$

$$\therefore$$ If $$f(x)=a^x$$, then $$f'(x)$$ = $$f'(0)$$ $$×f(x)$$.

• $$\therefore$$ rate of change of any exponential function is proportional to function itself.
• If $$a=e$$,  then we say that  $$\dfrac{d}{dx}e^x=e^x$$
• By chain rule,

$$\dfrac{d}{dx}e^u=e^x\dfrac{du}{dx}$$ where, $$u$$ is any function of $$x$$.

#### If $$f(x)=(2x^5-3x)\,e^x$$ , find $$f'(x)$$.

A $$e^x[2x^5+10x^4-3x-3]$$

B $$e^x[5x^5-x^4+x^3+3]$$

C $$e^x[5x^4-6x^3+8x+1]$$

D $$e^x[10x^5-4x^3+x+7]$$

×

$$f(x)=(2x^5-3x)e^x$$

$$\Rightarrow$$ $$f'(x)$$$$\underbrace{(2x^5-3x)\dfrac{d}{dx}(e^x)+e^x\dfrac{d}{dx}(2x^5-3x)}_{Product\,Rule}$$

$$=(2x^5-3x)e^x+e^x\,[10x^4-3]$$

$$=e^x[2x^5-3x+10x^4-3]$$

$$=e^x\,[2x^5+10x^4-3x-3]$$

### If $$f(x)=(2x^5-3x)\,e^x$$ , find $$f'(x)$$.

A

$$e^x[2x^5+10x^4-3x-3]$$

.

B

$$e^x[5x^5-x^4+x^3+3]$$

C

$$e^x[5x^4-6x^3+8x+1]$$

D

$$e^x[10x^5-4x^3+x+7]$$

Option A is Correct

# Finding the Inverse of a Function which contain Exponential Function in them

• To find the inverse of an function:
1. Let $$y = f(x)$$ be the given function.
2. Solve $$x$$ in terms of $$y$$.
3. Interchange $$x$$ and $$y$$, the new $$y$$ obtained is the required inverse.

#### Find the inverse function of the function  $$f(x)=\dfrac {e^x}{2+3e^x}$$.

A $$f^{-1}(x)=\ell n\,x$$

B $$f^{-1}(x)=\ell n\Bigg(\dfrac {2x}{1-3x}\Bigg)$$

C $$f^{-1}(x)=e^{x^2}$$

D $$f^{-1}(x)=\ell n\Bigg(\dfrac {5x}{1+x}\Bigg)$$

×

$$f(x)=\left (\dfrac {e^x}{2+3e^x}\right)$$

$$y=f(x)$$

$$\Rightarrow y=\left (\dfrac {e^x}{2+3e^x}\right)$$

$$2y+3y\,e^x=e^x$$

$$\Rightarrow e^x(1-3y)=2y$$

$$\Rightarrow e^x=\dfrac {2y}{1-3y}$$

$$\Rightarrow x=\ell n\;\dfrac {2y}{1-3y}$$ (take log on both sides to base e)

Interchange $$x$$ and $$y$$

$$y=\ell n\left (\dfrac {2x}{1-3x}\right)=f^{-1}(x)$$

### Find the inverse function of the function  $$f(x)=\dfrac {e^x}{2+3e^x}$$.

A

$$f^{-1}(x)=\ell n\,x$$

.

B

$$f^{-1}(x)=\ell n\Bigg(\dfrac {2x}{1-3x}\Bigg)$$

C

$$f^{-1}(x)=e^{x^2}$$

D

$$f^{-1}(x)=\ell n\Bigg(\dfrac {5x}{1+x}\Bigg)$$

Option B is Correct

#### Consider the graph of there functions on the same $$x-y$$ axis. Which of the following is the correct statement ?

A (1)  is the graph of $$f(x)=\left(\dfrac{1}{10}\right)^x=10^{-x}$$ (2)  is the graph of $$f(x)=\left(\dfrac{1}{9}\right)^x=9^{-x}$$ (3)  is the graph of $$f(x)=\left(\dfrac{1}{7}\right)^x=7^{-x}$$

B (1)  is the graph of $$f(x)=\left(\dfrac{1}{8}\right)^x=8^{-x}$$ (2)  is the graph of $$f(x)=\left(\dfrac{1}{10}\right)^x=10^{-x}$$ (3)  is the graph of $$f(x)=\left(\dfrac{1}{6}\right)^x=6^{-x}$$

C (1)  is the graph of $$f(x)=\left(\dfrac{1}{6}\right)^x=6^{-x}$$ (2)  is the graph of $$f(x)=\left(\dfrac{1}{2}\right)^x=2^{-x}$$ (3)  is the graph of $$f(x)=\left(\dfrac{1}{3}\right)^x=3^{-x}$$

D (1)  is the graph of $$f(x)=\left(\dfrac{1}{7}\right)^x=7^{-x}$$ (2)  is the graph of $$f(x)=\left(\dfrac{1}{5}\right)^x=5^{-x}$$ (3)  is the graph of $$f(x)=\left(\dfrac{1}{9}\right)^x=9^{-x}$$

×

(3)  is the steepest declining graph, then (2) and (1)

$$\therefore$$ 'a' should be the least for (3) and the greatest for (1)

Hence, option (A) is correct.

### Consider the graph of there functions on the same $$x-y$$ axis. Which of the following is the correct statement ?

A

(1)  is the graph of $$f(x)=\left(\dfrac{1}{10}\right)^x=10^{-x}$$

(2)  is the graph of $$f(x)=\left(\dfrac{1}{9}\right)^x=9^{-x}$$

(3)  is the graph of $$f(x)=\left(\dfrac{1}{7}\right)^x=7^{-x}$$

.

B

(1)  is the graph of $$f(x)=\left(\dfrac{1}{8}\right)^x=8^{-x}$$

(2)  is the graph of $$f(x)=\left(\dfrac{1}{10}\right)^x=10^{-x}$$

(3)  is the graph of $$f(x)=\left(\dfrac{1}{6}\right)^x=6^{-x}$$

C

(1)  is the graph of $$f(x)=\left(\dfrac{1}{6}\right)^x=6^{-x}$$

(2)  is the graph of $$f(x)=\left(\dfrac{1}{2}\right)^x=2^{-x}$$

(3)  is the graph of $$f(x)=\left(\dfrac{1}{3}\right)^x=3^{-x}$$

D

(1)  is the graph of $$f(x)=\left(\dfrac{1}{7}\right)^x=7^{-x}$$

(2)  is the graph of $$f(x)=\left(\dfrac{1}{5}\right)^x=5^{-x}$$

(3)  is the graph of $$f(x)=\left(\dfrac{1}{9}\right)^x=9^{-x}$$

Option A is Correct

# Limits of Exponential Function

• If $$a>1$$ then,

(1)  $$\lim\limits_{x\to \infty}\;a^x=\infty$$

(2)  $$\lim\limits_{x\to\, -\infty}\;a^x=0$$

• If $$0<a<1$$ then,

(1)  $$\lim\limits_{x\to \infty}\;a^x=0$$

(2)  $$\lim\limits_{x\to -\infty}\;a^x=\infty$$

$$x$$ axis will always be horizontal asymptote of the exponential function  $$y=a^x$$.

#### Find  $$\lim\limits_{x\to -\infty}\;\left(\dfrac{2^x-1}{3}\right)$$

A $$-\dfrac{1}{3}$$

B $$\infty$$

C $$\dfrac{1}3{}$$

D $$-\infty$$

×

$$\ell=\lim\limits_{x\to \,-\infty}\;\left(\dfrac{2^x-1}{3}\right)$$

$$=\dfrac{1}{3}\;\lim\limits_{x\to\, -\infty}\;\left({2^x-1}\right)$$

$$=\dfrac{1}{3}\left[\lim\limits_{x\to \,-\infty}2^x-\lim\limits_{x\to\,-\infty}-1\right]$$

Now,  $$\lim\limits_{x\to\,-\infty}\;2^x=0$$

$$\therefore\,\ell=\dfrac{1}{3}\left[\lim\limits_{x\to \,-\infty}\;2^x-1\right]$$

$$=\dfrac{1}{3}[0-1]$$

$$=-\dfrac{1}{3}$$

### Find  $$\lim\limits_{x\to -\infty}\;\left(\dfrac{2^x-1}{3}\right)$$

A

$$-\dfrac{1}{3}$$

.

B

$$\infty$$

C

$$\dfrac{1}3{}$$

D

$$-\infty$$

Option A is Correct