Learn derivative of e^x, differentiation of exponential functions, Practice equation Integral of exponential function to find the absolute maximum or minimum values & tangent Line calculus.
To differentiate the function which involves exponential function. We use the following:
(1) \(\dfrac{d}{dx}e^{f(x)}=e^{f(x)}× \)\(f'(x)\)
(2) Product rule \(\to \dfrac{d}{dx}(f(x)\,g(x))=f(x)\)\(g'(x)\)+\(f'(x)\)\(g(x)\)
(3) Chain rule \(\to \dfrac{d}{dx}(f(g(x)))=\)\(f'(g(x))\)×\(g'(x)\)
A \(e^{-3x}[5sin\,x+cos\,x]\)
B \(e^{5x}\)
C \(e^{-3x}[3cos\,2x-2sin\,2x]\)
D \(e^{-3x}[2cos\,2x-3sin\,2x]\)
A \(f\) is increasing in \((-\infty,\,-1)\) and decreasing in \((-1,\,0)\,\cup(0,\,\infty)\)
B \(f\) is increasing in \((-\infty,\,2)\) and decreasing in \((2,\,\infty)\)
C \(f\) is increasing in \((-\infty,\,0)\) and decreasing in \((0,\,\infty)\)
D \(f\) is increasing in \((-\infty,\,-3)\) and decreasing in \((-3,\,0)\,\cup(0,\,\infty)\)
\(\dfrac{d}{dx}(e^x)=e^x\)
\(\Rightarrow\int e^xdx=e^x+C\)
If the integral function is of the form \(\displaystyle I=\int \)\(f'(x)\)\(e^{f(x)}dx\) then we make the substitution .
\(f(x)=t\Rightarrow f'(x)dx=dt\)
\(\displaystyle I=\int e^tdt=e^t+C\)
\(\therefore\,I=e^{f(x)}+C\)
\(\displaystyle\therefore\,\int\,\)\(f'(x)\)\(e^{f(x)}dx=e^{f(x)}+C\)
So whenever we see \(e^{f(x)}\) term we think of the substitution \(y=f(x)\).
A \(-e^{cos\,x}+C\)
B \(e^{x^2}+C\)
C \(-e^{sin\,x}+C\)
D \(e^x+C\)
\(e\) is a number such that \(\lim\limits_{h\rightarrow0} \left ( \dfrac {e^h-1}{h} \right)=1\)
Consider,
\(\dfrac {d}{dx}(a^x)=\lim\limits_{h\rightarrow0} \left ( \dfrac {a^{x+h}-a^x}{h} \right)\)
\(=a^x\;\lim\limits_{h\rightarrow0} \left ( \dfrac {a^{h}-1}{h} \right)\)
\(\dfrac {d}{dx}(a^x)=a^x\;×\ell na \)
\( =a^x\;\ell na\)
If, \(a=e\) then we say that
\(\dfrac {d}{dx}\;e^x=e^x\,ln(e)\)
\(=e^x\)
\(y-f(a)=\)\(f'(a)\)\((x-a)\)
A \(x-y+1=0\)
B \(5x-y+7=0\)
C \(2x+y+3=0\)
D \(y-x-1=0\)
To find the absolute maximum or minimum values of a continuous function \(f\) in an interval [a,b], we do the following steps:
A Absolute maximum value \(=f(2)=\dfrac{4}{e^2}\), absolute minimum value \(=f(0)=0\)
B Absolute maximum value \(=f(-1)=e\), absolute minimum value \(=f(0)=0\)
C Absolute maximum value \(=f(2)=4e^2\), absolute minimum value \(=f(2)=\dfrac{4}{e^2}\)
D Absolute maximum value \(=f(3)=\dfrac{9}{e^3}\), absolute minimum value \(=f(4)=\dfrac{16}{e^4}\)
A \(\alpha=1,\;\alpha=-2\)
B \(\alpha=5,\;\alpha=1\)
C \(\alpha=2,\;\alpha=3\)
D \(\alpha=6,\;\alpha=-2\)
\(\therefore\) \(f'(0)\)\(=1\)
A \(2e^4\)
B \(2e^3\)
C \(\dfrac{3}{e}\)
D \(\dfrac{1}{e^2}\)
