Calculus 1

Find the inflection points and intervals of concavity for the following function

$y = \frac{x^2 + 1}{x^2}$

You have been asked to design a 1 liter can in the shape of a right circular cylinder. What dimensions use the least amount of material for the can? (minimize surface area)

Explain why the following limit can not be found using l'Hospital's Rule then find the limit using a different method.

$\lim_{x\rightarrow \infty} \frac{x + \cos{(x)}}{x}$

Evaluate the following limit

$\lim_{x\rightarrow 0} \frac{\sin{(3x)}}{\sin{(4x)}}$

Let $f(x) = \ln{(x)}$ Find the linearization of $f$ at $1$ and use it to evaluate $\ln{(0.9)}$

Suppose that a spherical container has a radius of $1 \pm 0.001 m$. Approximate the corresponding possible error in the calculated volume.

Find the linearization of $\csc{x}$ at $x = \frac{\pi}{4}$ and use it to approximate $\csc{1}$. Also find the error and percentage error.

Find the local linearization of $\ln{(x)}$ at $x = e^2$ and use it to approximate $\ln{(7.4)}$. Also find the error and percentage error.

Starting with an initial value $x_1 = 1$, perform 2 iterations of Newton's Method on $f(x) = x^3 - x - 1$ to approximate the root.

Use Newton's Method to approximate a solution to $2\cos{(x)} = 3x$ (Let $x_0 = \frac{\pi}{6}$ and find $x_2$)

Evaluate the following indefinite integral

$\displaystyle\int (\sqrt{x} - 5 \sqrt[3]{x^2}) \ dx$

Evaluate the indefinite integral

$\displaystyle\int (\frac{3}{x^2} - \frac{1}{x}) \ dx$

Evaluate the indefinite integral

$\int \frac{2x^5}{x^6 + 3}^2\ dx$

Evaluate the indefinite integral

$\displaystyle\int (x^4 - \frac{1}{3\sqrt{x}} + \frac{2}{5}x^{-\frac{4}{3}}) \ dx$

Evaluate the following integral

$\displaystyle\int\cos^4(x)\sin(x) \ dx$

Find $\displaystyle\int\sin(x)\sin(\cos{x}) \ dx$

Find $\displaystyle\int\frac{2x}{1 + 2x^2} \ dx$

Evaluate $\displaystyle\int\frac{e^{ \ p \ \tan^{-1}(x)}}{1 + x^2} \ dx$

Evaluate the definite integral below

$\displaystyle\int_{-2}^2 \ {x^2 \cos{(\frac{x^3}{8})}} \ dx$

Evaluate the following definite integral

$\displaystyle\int_0^4 \ x \sqrt{x^2 + 9} \ dx$