Calculus 1

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 $\ln{(0.9)}$

Find the linearization of $f(x) = {(1 + x)}^{P}$ at $0$, and approximate $f(\sqrt{0.99})$

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

Use Newton's method for approximating roots of functions to approximate $\sqrt{0.99}$

Approximate $\sqrt[4]{75}$ using the Newton Raphson method

Use linear approximations to estimate $\sqrt{8}$. Also find the error and percentage error.

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.

Approximate $f(x) = \sqrt[3]{x}$ at $x = 26$

Let $f(x) = \sqrt{x}$ at $x = 4$ and $\Delta{x} = 0.02$

Find $dx$, $dy$, $\Delta{y}$

Given a circle with a circumference of 56 inches with an error of $\pm{1.2}$ inches. Find the percent error of the area of the circle.

A 12 by 12 square is produced with an error of $\pm{\frac{1}{64}}$ inches in the length of each side. Find the the percent error in the area of one of these squares.

Use the Newton Raphson method to approximate the real zero close to $x = 1$ until two successive approximations differ by less than 0.005 for the following function

$f(x) = 2x^2 - 3$

Find the real zeros of f(x) and find the x intercepts for $f(x) = x^2 -5$

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 the solution to the following equation

$\cos{(x)} = \frac{x}{5}$

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

$\displaystyle\int{3x^6} dx$

Evaluate the indefinite integral

$\displaystyle\int 7\sqrt{x}\ dx$