# Tag Archives: Exponential Function \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 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 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 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 & […] # Exponential Equations using Logarithms We can find solutions to simple exponential equations where we can 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 same bases, such as 2^x=5. We use \textit{logarithms} to find the exact […] # Natural Exponential Graphs  \large 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 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. Example […] # Integration by Reverse Chain Rule By recalling the chain rule, the 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 and 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 to achieve a valid solution of the differential equations, such as the product, quotient, and chain rules.Many students missed applying the chain rule, resulting in an unexpected […]