MATH 144: Calculus for the Physical Sciences I

Subsection 1.3.1 Things to know

• Lines:
  • Given two points $P_1(x_1,y_1)$ and $P_2(x_2, y_2)$ in the $xy$-plane,
    $$\begin{align*} \Delta x =\amp x_2-x_1\qquad \text{is called the change in x, or the ``run'',}\\ \Delta y =\amp y_2-y_1\qquad \text{is called the change in y, or the ``rise''.} \end{align*}$$
    If $x_1\neq x_2\text{,}$ the line passing through the points $P_1$ and $P_2$ is non-vertical, and its slope is
    $$\begin{equation*} m=\frac{y_2-y_1}{x_2-x_1}=\frac{\Delta y}{\Delta x}=\frac{\text{rise}}{\text{run}} \end{equation*}$$
  • If $P_1 \neq P_2$ but $\Delta y = 0$ then $m=0$ and the line is horizontal. If $P_1 \neq P_2$ but $\Delta x = 0$ then the line is vertical, and its slope is undefined. A vertical line is not the graph of a function, since it does not pass the vertical line test.
  • The equation of a line has the form $y = m x + b\text{,}$ where $m$ is the slope and $b$ is the $y$-intercept.
  • Two lines $y = m_1 x + b_1$ and $y = m_2 x + b_2$ are parallel if $m_1 = m_2\text{.}$ They are perpendicular if $m_1 = - 1/m_2\text{.}$
  • To find the equation of the line with a given slope $m=a$ passing through a point $P(x_1,y_1)\text{,}$ we can use the point-slope formula:
    $$\begin{equation*} y-y_1 = m(x-x_1). \end{equation*}$$
    We expand and gather terms to get an equation of the form $y=mx + b\text{.}$
  • To find the equation of the line passing through two points $P_1(x_1,y_1)$ and $P_2(x_2,y_2)\text{,}$ we first determine that the slope of the line is
    $$\begin{equation*} m = \frac{y_2-y_1}{x_2-x_1}, \end{equation*}$$
    and then substitute into the point-slope formula to get
    $$\begin{equation*} y-y_1 = \frac{y_2-y_1}{x_2-x_1} (x-x_1). \end{equation*}$$
    We expand and gather terms to get an equation of the form $y=mx+b\text{.}$
  • The distance between two points $P_1(x_1,y_1)$ and $P_2(x_2, y_2)$ is
    $$\begin{equation*} d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}. \end{equation*}$$
• Triangles: The area of a triangle is $A = \frac{1}{2} b h\text{,}$ where $b$ is the base and $h$ the height.
• Circles:
  • The equation of a circle with centre $(h,k)$ and radius $r$ is
    $$\begin{equation*} (x-h)^2 + (y-k)^2 = r^2. \end{equation*}$$
  • The area of a circle is $A = \pi r^2\text{.}$
  • The circumference of a circle is $C = 2 \pi r\text{.}$
  • The area of a sector of a circle is $A = \frac{1}{2} \theta r^2$ where $\theta$ is the angle in radians.
  • The length of an arc is $L = \theta r\text{.}$
• Parabolas: A parabola is the graph of a function of the form $y = a x^2 + b x + c\text{.}$
• Ellipses: The equation of an ellipse with centre $(h,k)$ is
  $$\begin{equation*} \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1. \end{equation*}$$
• Hyperbolas: The equation of a hyperbola with “centre” $(h,k)$ is
  $$\begin{equation*} \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1. \end{equation*}$$
• Three-dimensional objects:
  • The volume of a sphere of radius $r$ is $V = \frac{4}{3} \pi r^3\text{.}$
  • The surface area of a sphere is $A = 4 \pi r^2\text{.}$
  • The volume of a cylinder is $V = \pi r^2 h$ where $r$ is the radius and $h$ the height.
  • The volume of a right circular cone is $V = \frac{1}{3} \pi r^2 h\text{.}$