Calculus > Mean Value Theorem The Mean Value Theorem states that if a function is continuous on a closed interval [a,b] and if the function is differentiable on the open interval (a,b) then there is a number c in (a,b)…

# Quadratic Approximation in Calculus: How to Use it, Step by Step

Calculus > Quadratic Approximation Quadratic polynomial approximations are specific examples of a useful class of quadratic approximations known as Taylor polynomials. The basic idea is that you want to approximate a function with a line. However, a straight line normally…

# How to use Taylor Polynomials to Approximate a Function

Calculus > Taylor polynomials can be used to approximate a function around any value for a differentiable function. Taylor polynomials look a little ugly, but if you break them down into small steps, it’s actually a fast way to approximate…

# Second Derivative Test: Definition, Finding Extrema

Calculus > Second Derivative Test The second derivative test is used in calculus to find intervals where a function has a relative maxima and minima. You can also use the second derivative test to determine concavity. The second derivative test…

# First Derivative Test: Step by Step

Calculus > The first derivative test is one way to study increasing and decreasing properties of functions. The test helps you to: Find the intervals where a function is decreasing or increasing. Identify local minima and maxima. Sketch a graph…

# Rolle’s Theorem: Definition and Calculating

Calculus > Rolle’s Theorem What is Rolle’s Theorem? Rolle’s theorem is a special case of the mean value theorem. Rolle’s theorem states that for any continuous, differentiable function that has two equal values at two distinct points, the function must…

# Linearization and Linear Approximation in Calculus

Calculus > Linearization and Linear Approximation in Calculus Linearization, or linear approximation, is just one way of approximating a tangent line at a certain point. Seeing as you need to take the derivative in order to get the tangent line,…

# Proof of the Derivative Tan x: Easy Steps

Calculus > Proof of the Derivative Tan x The derivative tan x is sec2x: There are a couple of ways to prove the derivative tan x. You could start with the definition of a derivative and prove the rule using…

# How to Find a Second Derivative

Calculus > The second derivative, f”, is the derivative of the derivative f’. In other words, in order to find a second derivative, take the derivative of the derivative. The second derivative is especially useful when it comes to classifying…

# How to Figure Out When a Function is Not Differentiable

Calculus > How to Figure Out When a Function is Not Differentiable If there’s no limit to the slope of the secant line (in other words, if the limit does not exists at that point), then the derivative will not…