Compute the values of $dy$ and $\mathrm{\Delta }y$ for the function $y=e^{3x}+3x$ given $x=0$ and $\mathrm{\Delta }x=dx=0.05$.

Round your answers to four decimal places, if required. You can use a calculator, spreadsheet, browser, etc. to calculate $dy$ and $\mathrm{\Delta }y$.
$$\begin{array}{l}dy=\text{ Number }\\ \mathrm{\Delta }y=\text{ Number }\end{array}$$

Step 1 ：Given the function $y=e^{3x}+3x$, we are asked to compute the values of $dy$ and $\mathrm{\Delta }y$ at $x=0$ with $\mathrm{\Delta }x=dx=0.05$.

Step 2 ：The differential $dy$ is given by the derivative of the function times $dx$, i.e., $dy=f^{\prime }(x)dx$.

Step 3 ：The increment $\mathrm{\Delta }y$ is given by the difference in the function's values at $x+\mathrm{\Delta }x$ and $x$, i.e., $\mathrm{\Delta }y=f(x+\mathrm{\Delta }x)-f(x)$.

Step 4 ：First, we need to find the derivative of the function $f(x)=e^{3x}+3x$. The derivative is $f^{\prime }(x)=3e^{3x}+3$.

Step 5 ：Substitute $x=0$ and $dx=0.05$ into the expression for $dy$ to find its value. The result is $dy=0.3$.

Step 6 ：Substitute $x=0$ and $\mathrm{\Delta }x=0.05$ into the expression for $\mathrm{\Delta }y$ to find its value. The result is $\mathrm{\Delta }y=0.3118$.

Step 7 ：Final Answer: $$\begin{array}{l}dy=\boxed{{\displaystyle 0.3}}\\ Deltay=\boxed{{\displaystyle 0.3118}}\end{array}$$