Calculus 1

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

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

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

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

Find the following indefinite integral

$\displaystyle\int 3x^6 + 5 + 7\sqrt{x}\ dx$

Find the antiderivative of $x^3$

Evaluate the following integral

$\displaystyle\int (3x + 5x^2) \ dx$

Let $f^\prime(x) = 2x$ what is the antiderivative, $f(x)$ ?

Evaluate the indefinite integral below

$\displaystyle\int (x^3 + 2x - 1) \ dx$

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

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