\[=\int\frac{7 \times 4}{-7}e^{7x} dx\] \[=\int\frac{-4}{7}e^{u}du\] \[=\frac{-4}{7}e^{u} + const\] \[=\frac{-4}{7}e^{-7x} + const\] 2.
\[N(t) = \frac{1050}{t^3}-220t + const\] t = 1 and N(1) = 6530 \[Const = 6530 - 1050 + 220\] \[const = 5700\] \[ N(t) = \frac{1050}{t^3} - 220t + 5700 \] 3.
#set function
f_a = function(x) {2*x -9}
#Find the area
area <- integrate(f_a, 4.5, 8.5)$value
area <- round(as.numeric(area))
print(area)## [1] 16#set function
f_1 = function(x) {x + 2}
f_2 = function(x) {x^2 -2*x -2}
#Find the area
area_1 <- integrate(f_1, -1, 4)
area_2 <- integrate(f_2, -1, 4)
net_area <- round((area_1$value - area_2$value),4)
print(net_area)## [1] 20.8333\[f'(x) = 1.875 - \frac{907.5}{x^2} = 0\] \[x^2 = \frac{907.5}{1.875}\] \[x=22\]
\[=\int ln(9x) \times x^6dx = \frac{1}{7}x^7 \times ln(9x) - \int \frac{1}{7}x^7 \times \frac{1}{x}dx \] \[=\frac{7}{49}x^7 \times ln(9x) - \frac{1}{49}x^7 + const \] \[=\frac{1}{49}x^7(7ln(9x)-1) + const \]
\[ f(x) = \frac{1}{6x} \] \[ \int_{1}^{e^6}\frac{1}{6x}dx = \frac{1}{6}ln(x)\Big|_1^{e^6} = 1 \]