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

# Tag Archives: Exponential Function

# Integration of Exponential Functions

The base formula of integrating exponential function is obtained from deriving $e^x$. $$ \begin{align} \displaystyle \dfrac{d}{dx}e^x &= e^x \\ e^x &= \int{e^x}dx \\ \therefore \int{e^x}dx &= e^x +c \\ \end{align} $$ This base formula is extended to the following general formula. $$ \begin{align} \displaystyle \dfrac{d}{dx}e^{ax+b} &= e^{ax+b} \times \dfrac{d}{dx}(ax+b) \\ &= e^{ax+b} \times a \\ […]

# 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 $f(x)$. $$\displaystyle \dfrac{d}{dx}e^x = e^x$$ $$\displaystyle \dfrac{d}{dx}e^{f(x)} = e^{f(x)} \times f'(x)$$ 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

Inequalities are worked in exactly the same way except that there is 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 […]

# Exponential Equations using Logarithms

We can find solutions to simple exponential equations where we could make equal bases and then equate exponents (indices). For example, $2^{x}=8$ can be written as $2^x = 2^3$. Therefore the solution is $x=3$. However, it is not always easy to make the bases the same such as $2^x=5$. In these situations, we use $\textit{logarithms}$ […]

# Natural Exponential Graphs

$$y=e^x$$ Natural Exponential Graphs also follow the rule of translations and transformations. Example 1 Sketch the graphs of $y=e^x$ and $y=-e^x$. Reflected to the $x$-axis. Example 2 Sketch the graphs of $y=e^x$ and $y=-e^{-x}$. Example 3 Sketch the graphs of $y=e^x$ and $y=e^{-x}$. Example 4 Sketch the graphs of $y=e^x$ and $y=e^{x+1}$. Translated to left. […]

# Exponential Graphs

Functions of the form exponential, where the base is a positive real number other than 1 are called exponential graphs or exponential functions.

# Integration by Reverse Chain Rule

By recalling the chain rule, Integration Reverse Chain Rule comes from the usual chain rule of differentiation. This skill is to be used to integrate composite functions such as \( e^{x^2+5x}, \cos{(x^3+x)}, \log_{e}{(4x^2+2x)} \). Let’s take a close look at the following example of applying the chain rule to differentiate, then reverse its order to […]

# Finding a Function from Differential Equation

The solution of a differential equation is to find an expression without \( \displaystyle \frac{d}{dx} \) notations using given conditions.Note that the proper rules must be in place in order to achieve the valid solution of the differential equations, such as product rule, quotient rule and chain rule particularly.Many students missed applying the chain rule […]