Tag Archives: Exponential Function

Definite Integral of Exponential Functions

Definite Integral of Exponential Functions

$$ \large \begin{align} \displaystyle\int_{n}^{m}{e^{ax+b}}dx &= \dfrac{1}{a}\big[e^{ax+b}\big]_{n}^{m} \\&= \dfrac{1}{a}\big[e^{am+b}-e^{an+b}\big] \\\end{align} $$ Example 1 Find $\displaystyle \int_{2}^{4}{e^{2x-4}}dx$, leaving the answer in exact form. \( \begin{align} \displaystyle\int_{2}^{4}{e^{2x-4}}dx &= \dfrac{1}{2}\big[e^{2x-4}\big]_{2}^{4} \\&= \dfrac{1}{2}\big[e^{2 \times 4-4}-e^{2 \times 2-4}\big] \\&= \dfrac{e^4-e^{0}}{2} \\&= \dfrac{e^4-1}{2} \end{align} \) Example 2 Find $\displaystyle \int_{0}^{1}{(e^x+1)^2}dx$. \( \begin{align} \displaystyle\int_{0}^{1}{(e^{2x} + 2e^x + 1)}dx &= \big[\dfrac{1}{2}e^{2x} + 2e^x + […]

Integration of Exponential Functions

Integration of Exponential Functions

Example 1 Find $\displaystyle \int{2e^x}dx$. $\displaystyle \int{2e^x}dx = 2e^x +c$ Example 2 Find $\displaystyle \int{e^{2x+1}}dx$. $\displaystyle \int{e^{2x+1}}dx = \dfrac{1}{2}e^{2x+1} + c$ Example 3 Find $\displaystyle \int{\dfrac{1}{e^x}}dx$. \( \begin{align} \displaystyle\int{\dfrac{1}{e^x}}dx &= \int{e^{-x}}dx \\&= \dfrac{1}{-1}e^{-x} +c \\&= -\dfrac{1}{e^x} +c\end{align} \) Example 4 Find $\displaystyle \int{\sqrt{e^x}}dx$. \( \begin{align} \displaystyle\int{\sqrt{e^x}}dx &= \int{(e^x)^{\frac{1}{2}}}dx \\&= \int{e^{\frac{x}{2}}}dx \\&= \dfrac{1}{2}e^{\frac{x}{2}} +c \\&= \dfrac{1}{2}\sqrt{e^x} […]

Derivative of Exponential Functions

Derivative of Exponential Functions

The functions $e^{-x}$, $e^{3x+2}$ and $e^{x^2+2x-1}$ are all of the form $e^{f(x)}$.$e^{f(x)} \gt 0$ for all $x$, no matter what the function $f(x)$.$$ \large \displaystyle \begin{align} \dfrac{d}{dx}e^x &= e^x \\\dfrac{d}{dx}e^{f(x)} &= e^{f(x)} \times f'(x) \end{align} $$ Example 1 Find $\displaystyle \dfrac{dy}{dx}$ if $y=e^{4x}$. \( \begin{align} \displaystyle\dfrac{dy}{dx} &= e^{4x} \times \dfrac{d}{dx}4x \\&= e^{4x} \times 4 \\&= […]

Exponential Inequalities using Logarithms

Exponential Inequalities using Logarithms

Inequalities worked in the same way, except there was a change of sign when dividing or multiplying both sides of the inequality by a negative number. \begin{array}{|c|c|c|} \hline\log_{2}{3}=1.6>0 & \log_{5}{3}=0.7>0 & \log_{10}{3}=0.5>0 \\ \hline\log_{2}{2}=1>0 & \log_{5}{2}=0.4>0 & \log_{10}{2}=0.3>0 \\ \hline\log_{2}{1}=0 & \log_{5}{1}=0 & \log_{10}{1}=0 \\ \hline\log_{2}{0.5}=-1<0 & \log_{5}{0.5}=-0.4<0 & \log_{10}{0.5}=-0.3<0 \\ \hline\log_{2}{0.1}=-3.3<0 & \log_{5}{0.1}=-1.4<0 & […]