slope

Examples of slope in the following topics:

• Slope

  • Slope is often denoted by the letter $m$.
  • In other words, a line with a slope of $-9$ is steeper than a line with a slope of $7$.
  • The slope of the line is $\frac{4}{5}$.
  • We can see the slope is decreasing, so be sure to look for a negative slope.
  • Calculate the slope of a line using "rise over run" and identify the role of slope in a linear equation

• Slope-Intercept Equations

  • Writing an equation in slope-intercept form is valuable since from the form it is easy to identify the slope and $y$-intercept.  
  • Let's write the equation $3x+2y=-4$ in slope-intercept form and identify the slope and $y$-intercept.
  • Now that the equation is in slope-intercept form, we see that the slope $m=-\frac{3}{2}$, and the $y$-intercept $b=-2$.
  • The value of the slope dictates where to place the next point.
  • The slope is $2$, and the $y$-intercept is $-1$.  

• Point-Slope Equations

  • The point-slope equation is another way to represent a line; only the slope and a single point are needed.
  • Given a slope, $m$, and a point $(x_{1}, y_{1})$, the point-slope equation is:
  • Example: Write the equation of a line in point-slope form, given a point $(2,1)$ and slope $-4$, and convert to slope-intercept form
  • Since we have two points, but no slope, we must first find the slope:
  • Plug this point and the calculated slope into the point-slope equation to get:

• Parallel and Perpendicular Lines

  • Recall that the slope-intercept form of an equation is: $y=mx+b$ and the point-slope form of an equation is: $y-y_{1}=m(x-x_{1})$, both contain information about the slope, namely the constant $m$.
  • This means that if the slope of one line is $m$, then the slope of its perpendicular line is $\frac{-1}{m}$.
  • The two slopes multiplied together must equal $-1$.
  • Also, the product of the slopes equals $-1$.
  • Again, start with the slope-intercept form and substitute the values, except the value for the slope will be the negative reciprocal.  

• Slope and Y-Intercept of a Linear Equation

  • For the linear equation y = a + bx, b = slope and a = y-intercept.
  • What is the y-intercept and what is the slope?
  • The slope is 15 (b = 15).
  • (a) If b > 0, the line slopes upward to the right.
  • (c) If b < 0, the line slopes downward to the right.

• Slope and Intercept

  • The concepts of slope and intercept are essential to understand in the context of graphing data.
  • The slope or gradient of a line describes its steepness, incline, or grade.
  • A higher slope value indicates a steeper incline.
  • Slope is normally described by the ratio of the "rise" divided by the "run" between two points on a line.
  • It also acts as a reference point for slopes and some graphs.

• The Derivative and Tangent Line Problem

  • The use of differentiation makes it possible to solve the tangent line problem by finding the slope $f'(a)$.
  • The slope of the secant line passing through $p$ and $q$ is equal to the difference quotient
  • If $k$ is known, the equation of the tangent line can be found in the point-slope form:
  • It barely touches the curve and shows the rate of change slope at the point.
  • Define a derivative as the slope of the tangent line to a point on a curve

• Derivatives and Rates of Change

  • Thus, to solve the tangent line problem, we need to find the slope of a line that is "touching" a given curve at a given point, or, in modern language, that has the same slope.
  • But what exactly do we mean by "slope" for a curve?
  • In this case, $y = f(x) = m x + b$, for real numbers m and b, and the slope m is given by:
  • This gives an exact value for the slope of a straight line.
  • If $x$ and $y$ are real numbers, and if the graph of $y$ is plotted against $x$, the derivative measures the slope of this graph at each point.

• Direction Fields and Euler's Method

  • Direction fields, also known as slope fields, are graphical representations of the solution to a first order differential equation.
  • The slope field is traditionally defined for differential equations of the following form:
  • An isocline (a series of lines with the same slope) is often used to supplement the slope field.
  • Then, from the differential equation, the slope to the curve at $A_0$ can be computed, and thus, the tangent line.
  • Along this small step, the slope does not change too much $A_1$ will be close to the curve.

• Demand Curve

  • The demand curve in a monopolistic competitive market slopes downward, which has several important implications for firms in this market.
  • The demand curve of a monopolistic competitive market slopes downward.
  • The demand curve for an individual firm is downward sloping in monopolistic competition, in contrast to perfect competition where the firm's individual demand curve is perfectly elastic.
  • Because the individual firm's demand curve is downward sloping, reflecting market power, the price these firms will charge will exceed their marginal costs.
  • As you can see from this chart, the demand curve (marked in red) slopes downward, signifying elastic demand.