By its nature, Calculus can be intimidating. But you can take some of the fear of studying Calculus away by understanding its basic principles, such as derivatives and antiderivatives, integration, and solving compound functions. Also discover a few basic rules applied to Calculus like Cramer’s Rule, and the Constant Multiple Rule, and a few others, and you’ll be on your way to acing the course.

AP Calculus AB – “Related Rates” Solving Related Rates Problems 1. Name the variables and constants. Determine what you are asked to find. Write an equation(s) that relates the variables. Implicitly differentiate with respect to time. Evaluate and solve for the unknown. Problem: two roads cross at right angles. An observer stands on the road 80 meters north of the intersection. She is watching a car traveling at 50m/sec due east. At how many meters per second is the car moving away from the observer 3 seconds after it passes through the intersection? I have: dv/dt - 50m/sec z = root(230) at 3 seconds.

The Most Important Derivatives and Antiderivatives to Know

The table below shows you how to differentiate and integrate 18 of the most common functions. As you can see, integration reverses differentiation, returning the function to its original state, up to a constant C.

The Riemann Sum Formula For the Definite Integral

The Riemann Sum formula provides a precise definition of the definite integral as the limit of an infinite series. The Riemann Sum formula is as follows:

Below are the steps for approximating an integral using six rectangles:

1. Increase the number of rectangles (n) to create a better approximation:

2. Simplify this formula by factoring out w from each term:

3. Use the summation symbol to make this formula even more compact:

   The value w is the width of each rectangle:

   Each h valueis the height of a different rectangle:

   Crazy car stuntsgamefort. So here is the Riemann Sum formula for approximating an integral using n rectangles:

4. For a better approximation, use the limit

5. to allow the number of rectangles to approach infinity:

Integration by Parts with the DI-agonal Method

The DI-agonal method is basically integration by parts with a chart that helps you organize information. This method is especially useful when you need to integrate by parts more than once to solve a problem. Use the following table for integration by parts using the DI-agonal method:

The Sum Rule, the Constant Multiple Rule, and the Power Rule for Integration

When you perform integration, there are three important rules that you need to know: the Sum Rule, the Constant Multiple Rule, and the Power Rule.

The Sum Rule for Integration tells you that it’s okay to integrate long expressions term by term. Here it is formally:

The Constant Multiple Rule for Integration tells you that it’s okay to move a constant outside of an integral before you integrate. Here it is expressed in symbols:

The Power Rule for Integration allows you to integrate any real power of x (except –1). Here’s the Power Rule expressed formally:

where n ≠ –1

How to Solve Integrals with Variable Substitution

In Calculus, you can use variable substitution to evaluate a complex integral. Variable substitution allows you to integrate when the Sum Rule, Constant Multiple Rule, and Power Rule don’t work.

1. Declare a variable u,set it equal to an algebraic expression that appears in the integral, and then substitute u for this expression in the integral.

2. Differentiate u to find

   and then isolate all x variables on one side of the equal sign.

3. Make another substitution to change dx and all other occurrences of x in the integral to an expression that includes du.

4. Integrate by using u as your new variable of integration.

5. Express this answer in terms of x.

How to Use Integration by Parts

When doing Calculus, the formula for integration by parts gives you the option to break down the product of two functions to its factors and integrate it in an altered form. To use integration by parts in Calculus, follow these steps:

1. Decompose the entire integral (including dx) into two factors.

2. Let the factor without dx equal u and the factor with dx equal dv.

3. Differentiate u to find du, and integrate dv to find v.

4. Use the formula:

5. Evaluate the right side of this equation to solve the integral.

How to Solve Compound Functions Where the Inner Function Is ax + b

Some integrals of compound functions f(g(x)) are easy to do quickly in Calculus. These include compound functions for which you know how to integrate the outer function f, and the inner function g(x) is of the form ax + b — that is, it differentiates to a constant.

6.2 Related Ratesap Calculus Worksheet

Here are some examples:

Solve Compound Functions Where the Inner Function Is ax

When figuring Calculus problems, some integrals of compound functions f(g(x)) are easy to do quickly. These include compound functions for which you know how to integrate the outer function f, and the inner function g(x) is of the form ax — that is, it differentiates to a constant.

Here are some examples:

This video lesson explores the concept of Related Rates, which is the study of what is happening over time.

To solve problems with Related Rates, we will need to know how to differentiate implicitly, as most problems will be formulas of one or more variables.

But this time we are going to take the derivative with respect to time, t, so this means we will multiply by a differential for the derivative of every variable!

But don’t let this confused you, as there are only four basic steps when problem-solving with Related Rates:

1. Write down every given information or quantity, even what you’re looking for, and make a sketch.
2. Determine the equation needed to solve for your unknown quantity.
3. Take the derivative implicitly, not forgetting to multiply by your differential for every single variable!
4. Plug in all of your given information or quantities and solve.

And here’s a big hint…

…most of the time all we have to do is find an equation that relates the rate we’re looking for to a rate that we already know, as Paul’s Online Notes so nicely states.

This means we need to look for geometric shapes, known formulas, and ratios.

6.2 Related Ratesap Calculus Answers

Ladder Sliding Down a Wall

Together we will solve these 8 classic questions:

• Pebble Dropped In Pond
• Air Being Pumped Into A Balloon
• Edges Of Cube Expanding
• Radar Tracking Station & Airplane Flight Path
• Sand Falling onto a Conical Pile
• Water Pouring into a Conical Tank
• Boat Pulled into a Dock
• Ladder Sliding Down a Wall

How to Solve Related Rates – Video

1 hr 35 min

• Overview of Related Rates + Tips to Solve Them
• 00:02:58 – Increasing Area of a Circle
• 00:12:30 – Expanding Volume of a Sphere
• 00:21:15 – Expanding Volume of a Cube
• 00:26:32 – Calculate the Speed of an Airplane
• 00:39:13 – Conical Sand Pile
• 00:51:19 – Conical Water Tank
• 00:59:59 – Boat & Winch
• 01:09:13 – Ladder Sliding Down A Wall

Related Rates

Related Rates – 2 Examples

Get access to 6 more examples and over 150 HD videos with your subscription

Monthly, Half-Yearly, and Yearly Plans Available

Not yet ready to subscribe? Take Calcworkshop for a spin with our FREE limits course