Question

# 1.) A wire 20 m long is cut into two pieces. One is bent into an equilateral triangle other into a circle. How should the wire be cut so that the total resulting area is a a.)maximum? b.) minimum? Give an exact answer. 2.) An aluminum can with a top is to constructed so that it has a volume of 16 cm³.How should it be constructed so it uses the least amount of aluminum? State the base radius and the height. \text { 3.) Evaluate } \int_{1}^{3}\left(2 x^{2}-x+1\right) d x \text { using the Riemann sum. Show your work for this one. } \text { 6.) } \int \sin 3 \theta \cos 3 \theta d \theta \text { 7.) } \int_{0}^{2} \frac{d x}{x-1} \text { 8.) } \int 3 x^{5} \sqrt{x^{2}+1} d x \text { 9.) } \int_{1}^{4} \frac{x^{2}-3 x+1}{\sqrt{x}} d x \text { 10.) } \int \csc ^{2} x \sin (\cot x) d x \text { 11.) } \int_{-\pi / 4}^{\pi / 4} \frac{\tan x d x}{x^{4}+1} \text { 12.) } \int \frac{\sin (1 / x) d x}{x^{2}} \text { 13.) } \int_{1}^{2} \frac{x+1}{\sqrt{x^{2}+2 x}} d x \text { 14.) } \int\left(x+x^{-1}\right)^{2} d x \text { 15.) } \int \frac{d x}{4 \sec x} \text { 16.) } \int\left(\sqrt{x^{2}}-\sin x\right) d x \text { If } g(t)=\int_{x^{2}}^{\tan x} \sqrt{t^{3}+6} d t \text {, find } g^{\prime}(x) \text { If } f^{\prime \prime}(x)=6 x+5, f(0)=1, f(1)=3 \text {, find } f(x)  Fig: 1  Fig: 2  Fig: 3  Fig: 4  Fig: 5  Fig: 6  Fig: 7  Fig: 8  Fig: 9  Fig: 10  Fig: 11  Fig: 12  Fig: 13  Fig: 14  Fig: 15  Fig: 16  Fig: 17