Calculus Related Rates Problem: At what rate does the angle change as a ladder slides away from a house? A 10-ft ladder leans against a house on flat ground. The house is to the left of the ladder. The base of the ladder starts to slide away from the house at 2 ft/s. At what… [read more]

Calculus Related Rates Problem: As a snowball melts, how fast is its radius changing? A spherical snowball melts at the rate of $2 \pi$ cm$^3$/hr. It melts symmetrically such that it is always a sphere. How fast is its radius changing at the instant $r = 10$ cm? Hint: The volume of a sphere is… [read more]

In Part 1 of this series, we illustrated three of the most common tactics you must know to use in order to be able to solve limit problems in Calculus: (See Part 1 for details on those.) In this post, we’re going to look at two other tactics you’ll frequently need to invoke. I. Tactic… [read more]

If you’re like many Calculus students, you understand the idea of limits, but may be having trouble solving limit problems in your homework, especially when you initially find “0 divided by 0.” In this post, we’ll show you the techniques you must know in order to solve these types of problems. I. The idea of… [read more]

Handy Table of Integrals Power of x \begin{align*} \int a \, dx &= ax + C \\ \\ \int x^n \, dx &= \frac{x^{n+1}}{n+1} + C \qquad n \ne -1 \\ \\ \int \frac{1}{x} \, dx &= \ln |x| + C \\ \end{align*} Exponential and Logarithmic \begin{align*} \int e^x \, dx &= e^x + C… [read more]

Handy Table of Derivatives Want lots of examples to see how to calculate derivatives? Visit our free Calculating Derivatives: Problems & Solutions page! Power of x \begin{align*} \frac{d}{dx} \left(c\right) &= 0 \\[8px] \frac{d}{dx} \left(cx\right) &= c \\[8px] \frac{d}{dx} \left(cx^n\right) &= ncx^{n-1} \\[8px] \end{align*} For example, \[\dfrac{d}{dx}5x^3 = 3 \cdot 5x^2 = 15x^2 \] You’ll also… [read more]

Differentiation Rules Here’s a handy summary of the differentiation rules you’ll frequently use. Product Rule The differentiation rule for the product of two functions: \begin{align*} (fg)’&= f’g + fg’\\[8px] &= [{\small \text{ (deriv of the 1st) } \times \text{ (the 2nd) }}]\, + \,[{\small \text{ (the 1st) } \times \text{ (deriv of the 2nd)}}] \end{align*}… [read more]

Handy Table of Trig Function Derivatives Want lots of examples to see how to calculate derivatives? Visit our free Calculating Derivatives: Problems & Solutions page! \begin{align*} \frac{d}{dx}\left(\sin x\right) &= \cos x \\[8px] \dfrac{d}{dx}\left(\cos x\right) &= -\sin x \\[8px] \dfrac{d}{dx}\left(\tan x\right) &= \sec^2 x \\[8px] \frac{d}{dx}\left(\csc x\right) &= -\csc x \cot x \\[8px] \frac{d}{dx}\left(\sec x\right) &=… [read more]

A student recently wrote to ask if we’d help solve a common related rates problem about water draining from a cylindrical tank. The problem was something like this: Cylinder Drains Water. A cylinder filled with water has a 3.0-foot radius and 10-foot height. It is drained such that the depth of the water is decreasing… [read more]

Are you having trouble solving Calculus limit problems, even though you understand the concept? In this post we explain three approaches you’ll use again and again, especially in problems where you initially get “0/0.” I. Factoring You’ll use this approach most often. For example, consider the problem $$\lim_{x \to 4}\dfrac{x^2 – 16}{x-4} = ?$$ If… [read more]