In the case of a road, the "rise" is the change in altitude, while the "run" is the difference in distance between two fixed points, as long as the distance for the measurement is not large enough that the earth's curvature should be considered as a factor. Slope is essentially the change in height over the change in horizontal distance, and is often referred to as "rise over run." It has applications in gradients in geography as well as civil engineering, such as the building of roads. A vertical line has an undefined slope, since it would result in a fraction with 0 as the denominator.A line has a constant slope, and is horizontal when m = 0.A line is decreasing, and goes downwards from left to right when m A line is increasing, and goes upwards from left to right when m > 0.Given m, it is possible to determine the direction of the line that m describes based on its sign and value: The larger the value is, the steeper the line. Generally, a line's steepness is measured by the absolute value of its slope, m. Since now we have the slope of this line, and also the coordinates of a point on the line, we can get the whole equation of this tangent line.Slope, sometimes referred to as gradient in mathematics, is a number that measures the steepness and direction of a line, or a section of a line connecting two points, and is usually denoted by m. The slope of the tangent line at this point of tangency, say “\(a\)”, is the instantaneous rate of change at \(x=a\) (which we can get by taking the derivative of the curve and plugging in “\(a\)” for “\(x\)”). Let’s revisit the equation of a tangent line, which is a line that touches a curve at a point but doesn’t go through it near that point. We will talk about the Equation of a Tangent Line with Implicit Differentiation here in the Implicit Differentiation and Related Rates section. Also, there are some Tangent Line Equation problems using the Chain Rule here in The Chain Rule section. Note that we visited Equation of a Tangent Line here in the Definition of the Derivative section. Applications of Integration: Area and Volume.Exponential and Logarithmic Integration.Riemann Sums and Area by Limit Definition.Differential Equations and Slope Fields.Antiderivatives and Indefinite Integration, including Trig Integration.Derivatives and Integrals of Inverse Trig Functions.Exponential and Logarithmic Differentiation.Differentials, Linear Approximation, and Error Propagation.Curve Sketching, including Rolle’s Theorem and Mean Value Theorem.Implicit Differentiation and Related Rates.Equation of the Tangent Line, Tangent Line Approximation, and Rates of Change.Basic Differentiation Rules: Constant, Power, Product, Quotient, and Trig Rules.Differential Calculus Quick Study Guide.Polar Coordinates, Equations, and Graphs.Law of Sines and Cosines, and Areas of Triangles.Linear and Angular Speeds, Area of Sectors, and Length of Arcs.Conics: Circles, Parabolas, Ellipses, and Hyperbolas.Graphing and Finding Roots of Polynomial Functions.Graphing Rational Functions, including Asymptotes.
Introduction to the Graphing Display Calculator (GDC).Direct, Inverse, Joint and Combined Variation.Coordinate System and Graphing Lines, including Inequalities.Types of Numbers and Algebraic Properties.Introduction to Statistics and Probability.Powers, Exponents, Radicals (Roots), and Scientific Notation.Multiplying and Dividing, including GCF and LCM.