Definition of Differentiable Differentiable means:  the function has derivative the derivative f’(x) must exist for every value in the function’s domain Examples The use of differentiable function When a function is differentiable, we can use all the power of calculus when working with it. When a function is differentiable, it is continuous. Differentiable ⇒ Continuous … Continue reading "Differentiable"

Definition Integration by Substitution (also called “The Reverse Chain Rule” or “u-Substitution” ) is a method to find an integral, but only when it can be set up in a special way. Examples f(x) dx u du Substitution Integral of ∫ f(u) du Integral of ∫ f(x) dx ∫ (x+1)3 dx x+1 1 dx ∫ u3 du ∫ … Continue reading "Integration by Substitution (Reverse Chain Rule)"

Definition Integration is used to find areas, volumes, central points and often is used to find the area underneath the graph of a function. Common Functions of Integration   Function Integral Constant ∫ a dx ax + c Variable ∫ x dx Square ∫ x2 dx   Reciprocal ∫ (1/x) dx ln|x| + c Exponential ∫ … Continue reading "Integration"

Derivative of a function of a real variable measures the sensitivity to change of the function value (output) with respect to a change in its argument (input). Derivatives are a fundamental tool of calculus. The derivative of a function of a single variable at a chosen input value (when it exists) is the slope of the tangent … Continue reading "Common Derivatives"

A Differential Equation is an equation with a function and one or more of its derivatives: Example: an equation with the function y and its derivative Homogeneous Differential Equations A first order Differential Equation is Homogeneous when it can be in this form: We can solve it using Separation of Variables but first we create … Continue reading "Homogeneous Differential Equations"

When you can’t solve for y, find the derivative Implicit vs Explicit A function can be explicit or implicit: Explicit: y = some function of x When we know x we can calculate y directly Implicit: some function of y and x equals something else Knowing x does not lead directly to y Example How … Continue reading "Implicit Differentiation"

Calculus is the mathematical study of continuous change. The 2 branches are related to each other by the fundamental theorem of calculus. Both branches make use of the fundamental notions of convergence of infinite series & infinite sequences to a well-defined limit. Derivatives (Differential) Differentiable Common Derivatives Implicit Differentiation Homogeneous Differential Equations Integration (Integral) Integration … Continue reading "Calculus"