Practice Limits of Exponential & Graphs of Exponential Function & e Calculus, finding the inverse of a function which contain exponential function in them.
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 \(f(x)=2\left(5^x\right)\)
\((0,\,1)\) is a point on all the graphs of form \(f(x)=a^x\) \((0<a<1)\)
\(f(x)=a^x\) pass through \((0,\,1)\) as \(a°=1\) \(\forall\;'a'\).
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 \((1.2)^3=1.728\)
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 \(f(x)=5^x\)
\(e\) is a number such that
\(\lim\limits_{h\to 0}\left(\dfrac{e^h-1}{h}\right)=1\)
\(\dfrac{d}{dx}(a^x)=\lim\limits_{h\to 0}\left(\dfrac{a^{x+h}-a^x}{h}\right)\)
\(=a^x\,\lim\limits_{h\to 0}\left(\dfrac{a^h-1}{h}\right)\)
\(\therefore\) If \(f(x)=a^x\), then \(f'(x)\) = \(f'(0)\) \(×f(x)\).
\(\dfrac{d}{dx}e^u=e^x\dfrac{du}{dx}\) where, \(u\) is any function of \(x\).
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 \(e^x[10x^5-4x^3+x+7]\)
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 \(f^{-1}(x)=\ell n\Bigg(\dfrac {5x}{1+x}\Bigg)\)
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 \(f(x)=\left(\dfrac{1}{9}\right)^x=9^{-x}\)
(1) \(\lim\limits_{x\to \infty}\;a^x=\infty\)
(2) \(\lim\limits_{x\to\, -\infty}\;a^x=0\)
(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 \(y=a^x\).
A \(-\dfrac{1}{3}\)
B \(\infty\)
C \(\dfrac{1}3{}\)
D \(-\infty\)