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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 \(a\).
  • \(a^{-n}=\dfrac{1}{a^n}\)  \(n\,\varepsilon N\)
  • \(a^{p/q}=\left(a^{1/q}\right)^p\)?

Illustration Questions

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 \(\Rightarrow \,a>1.\)

 

image

From the graph

 \(\to f(0)=3\)

\(\Rightarrow\,3=k×a^0\)

\(\Rightarrow\,k=3\)

image

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)

image

\(\therefore\,f(x)=3×4^x\)

image

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

image
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 \(0<a<1\)

 

Illustration Questions

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.

image

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

image

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

image

Consider the graph of there functions on the same \(x-y\) axis. Which of the following is the correct statement ?

image
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

Illustration Questions

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 ?

image
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  \(y=a^x\).

Illustration Questions

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

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 \(x\).

Illustration Questions

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.

Illustration Questions

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

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