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$$ \begin{align} \displaystyle \int_{n}^{m}{e^{ax+b}}dx &= \dfrac{1}{a}\big[e^{ax+b}\big]_{n}^{m} \\ &= \dfrac{1}{a}\big[e^{am+b}-e^{an+b}\big] \\ \end{align} $$ Example 1 Find $\displaystyle \int_{2}^{4}{e^{2x-4}}dx$, leaving the answer in exact form. \( \begin{align} \displaystyle \int_{2}^{4}{e^{2x-4}}dx &= \dfrac{1}{2}\big[e^{2x-4}\big]_{2}^{4} \\ &= \dfrac{1}{2}\big[e^{2 \times 4 – 4} – e^{2 \times 2 – 4}\big] \\ &= \dfrac{e^4 – e^{0}}{2} \\ &= \dfrac{e^4 – 1}{2} \\ \end{align} \) […]
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. $$ \begin{align} \displaystyle \dfrac{d}{dx}e^{ax+b} &= e^{ax+b} \times \dfrac{d}{dx}(ax+b) \\ &= e^{ax+b} \times a \\ […]
The functions $e^{-x}$, $e^{3x+2}$ and $e^{x^2+2x-1}$ are all of the form $e^{f(x)}$. $e^{f(x)} \gt 0$ for all $x$, no matter what the function $f(x)$. $$\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 \\ […]
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 […]
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 $\textit{logarithms}$ […]
$$y=e^x$$ 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. […]
Functions of the form exponential, where the base is a positive real number other than 1 are called exponential graphs or exponential functions.
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 […]
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 […]