Data 605 - Homework #13

\[\begin{split} \int{4e^{-7x}dx} &= \int{\frac{-7 \times 4}{-7}e^{-7x}dx} \\ &= \int{\frac{-4}{7}e^u du} \\ &= \frac{-4}{7}e^u+constant \\ &= -\frac{4}{7}e^{-7x}+ constant \end{split}\] \[\begin{split} N(t) &= \frac{1050}{t^3}-220t+C \\ N(1) &= 6530 \\ \frac{1050}{1^3}-220\times 1 +C &= 6530 \\ C &= 6530 - 1050 + 220 \\ C &= 5700 \end{split}\]

#Find area in-build function
f3 = function(x) {2*x -9}

#Find the difference between areas under the curve
area3 <- integrate(f3, 4.5, 8.5)$value
area3 <- round(as.numeric(area3))
print(area3)

## [1] 16

#Find area in-build function
f1 = function(x) {x + 2}
f2 = function(x) {x^2 -2*x -2}

#Find the difference between areas under the curve
area1 <- integrate(f1, -1, 4)
area2 <- integrate(f2, -1, 4)
area <- round((area1$value - area2$value),4)
print(area)

## [1] 20.8333

$$\begin{array}{rl}{f}^{\prime }(x)& =1.875-<!--\; ???\; -->\frac{907.5}{x^2}\\ f^{\prime }(x)& =0\\ 1.875-<!--\; ???\; -->\frac{907.5}{x^2}& =0\\ 1.875& =\frac{907.5}{x^2}\\ 1.875x^2& =907.5\\ x^2& =\frac{907.5}{1.875}\\ x& =\sqrt{\frac{907.5}{1.875}}\\ x& =\sqrt{484}\\ x& =22\end{array}$$ $$\begin{array}{rl}\int <!--\; ???\; -->ln(9x)\times <!--\; \times \; -->x^{6}dx& =\frac{1}{7}{x}^7\times <!--\; \times \; -->ln(9x)-<!--\; ???\; -->\int <!--\; ???\; -->\frac{1}{7}{x}^7\times <!--\; \times \; -->\frac{1}{x}dx\\ & =\frac{1}{7}{x}^7\times <!--\; \times \; -->ln(9x)-<!--\; ???\; -->\int <!--\; ???\; -->\frac{1}{7}{x}^{6}dx\\ & =\frac{7}{49}{x}^7\times <!--\; \times \; -->ln(9x)-<!--\; ???\; -->\frac{1}{49}{x}^7+constant\\ & =\frac{1}{49}{x}^7(7ln(9x)-<!--\; ???\; -->1)+constant\end{array}$$ $$\begin{array}{rl}{\int <!--\; ???\; -->}_1^{e^6}\frac{1}{6x}dx& =\frac{1}{6}ln(x){|}_1^{e^6}\\ & =\frac{1}{6}ln(e^6)-<!--\; ???\; -->\frac{1}{6}ln(1)\\ & =\frac{1}{6}\times <!--\; \times \; -->6-<!--\; ???\; -->\frac{1}{6}\times <!--\; \times \; -->0\\ & =1\end{array}$$