What is Calculus?
Technically, calculus is the study of rates of change. However, if you’ve never taken calculus before, “rates of change” might have too much meaning to you. Calculus is actually two separate categories: differentiation and integration. Differentiation is a tool where you can find an object’s velocity and acceleration based on the formula for that object’s position. Likewise, if you know the object’s velocity you can find that object’s acceleration. With integration, the opposite is true: you can find an object’s position if you know the object’s velocity or acceleration. This is illustrated in the following diagram.
You probably already have a good idea of what calculus is about if you’ve studied algebra. In algebra, you found the slope of a line using the slope formula, slope = rise/run. In calculus, you’ll be studying the slope of a curve. The slope of a curve isn’t as easy to calculate as the slope of a line, because the slope is different at every point of the curve (and there are technically an infinite amount of points on the curve!).
That’s where differentiation comes in: differentiation is a set of tools that allows you to find the slope of a tangent line at any point on any curve. This slope is called a derivative.
You can calculate the slope of a line at any point on a line by using two points (a and b on the top left picture) and the slope formula. However, you can’t use the same formula to calculate the slope of a point on a curve. Points a and b on the top right picture shows that the two points have very different tangent lines (shown in red). In order to calculate the slope of the tangent line at these points, you need calculus.
Basic Principles & Pre-Calc.
- The Intermediate Value Theorem.
- How to Read Symbols and Equations in Calculus.
- Solving the Quadratic Formula.
- How to Find the Domain and Range of a Function.
- Vertical Line Test for a Function.
General Info & Problem Solving
- How to find acceleration
- AP Calculus Exam: How to Justify Answers.
- Bessel Functions.
- How to Find Critical Numbers
- How to Eliminate Exponents
- How to Find Intercepts
- Figuring out Function Composition
- How to Graph Transformations.
- How to Calculate Instantaneous Velocity
- How to Find the Intersection of Two Lines
- How to Find Maximum Profit in Calculus
- How to Find Minimum Profit
- How to Use Newton’s Method to Find x-intercepts of Functions
- Optimization Problems.
- How to describe the path of a baseball.
- Related Rates
- How to find Total Distance in Calculus
- How to find Velocity
- How to Calculate the Velocity of a Falling Object.
- How to Find the Vertical Tangent.
- How to calculate the volume of an egg using integral calculus.
- Calculus Word Problems.
Excel for Calculus
- Derivatives & Derivative Rules: How To Find the Derivative
- How to use Taylor Polynomials to Approximate a Function.
- How to Use the Quadratic Approximation in Calculus.
- How to Use the Mean Value Theorem.
- How to Use The First Derivative Test.
- How to Use the Second Derivative Test.
- How to use Rolle’s Theorem.
- How to Figure Out Linearization and Linear Approximation in Calculus.
- How to Use Implicit Differentiation.
- How to Write a Proof of the Derivative Tan x.
- How to Show a Proof of the Power Rule.
- How to Find a Second Derivative.
- How to Figure Out When a Function is Not Differentiable.
- How to Find a Derivative Using the Derivative Formula.
The Power Rule
- Differentiation in Calculus: Power Rule Multiplied by a Constant.
- How to Differentiate Exponents
- The General Power Rule.
- Indefinite Integral Rules
- How to integrate y with respect to x in calculus.
- How to Solve Improper Integrals
- How to Distinguish Between Proper and Improper Integrals
- How to Use U-Substitution for Definite Integrals
- How to Use U-Substitution for Trigonometric Functions
- How to prove the volume of a cone using integration in Calculus
- How to Find the Area Between Two Curves in Calculus
- How to Calculate Definite Integrals
- Riemann Sums.
- How to Use Sigma Notation to Evaluate Sums of Rectangular Areas
- How to Use The Fundamental Theorem of Calculus
- How to Integrate Using Integration by Substitution
- How to Use U Substitution to Find Integrals for Exponential Functions
- How to take the Integral of Natural Log
- How to Integrate using Integration by Parts
- How to Find the Antiderivatives (indefinite integrals) of power functions
You’re going to need to be familiar with trigonometric identities (or at least know where to look for them). Trigonometry is an entire semester-long class (sometimes two!), so it isn’t possible to put all of the identities here. But some identities show up a lot more frequently than others. These are the trigonometric identities you’ll use over and over again.
Tangent and Cotangent Identities
Other Trigonometric Identities
What are the Trigonometric Identities Used for in Calculus?
The above trigonometric identities will be used over and over again in your classes. They especially come in handy when it comes to figuring out derivatives or simplifying functions. For example, if you have a sin2 and a cos2 close to each other in a function, you might be able to cancel them out using a trigonometric identity. If you have a messy looking function with sin/cos/-cos2/sec and other components, look for ways to convert to sin or cos using the above trigonometric identities.
Other trigonometric functions
There are dozens of other possible trigonometric functions like arccosine, arctangent and arcsine, but the reality is you’ll rarely, or never use them. In the five semesters of calc I took in college (calc I/calc 2/advanced calc I/advanced calc II and one semester of calc-based statistics), I think I used the rarer trigonometric functions only once or twice. You’ll probably never see them on a test (they might be assigned as a “tricky” homework problem). If they are on a test, your instructor will (or should) provide a list of lesser-known trigonometric identities. As long as you know (and can use) the above identities, you should be all set for your class.
Series Convergence Tests
Often, you’ll want to know whether a series converges (i.e. reaches a certain number) or diverges (does not converge). Figuring this out can be an extremely difficult task — something that’s beyond the scope of even a calculus II course. Thankfully, mathematicians before you have calculated the convergence or divergence of many common series. This enables you to figure out whether a particular series may or may not converge.
Series Convergence Tests in Alphabetical Order
Alternating Series Convergence Tests
If for all n,an is positive, non-increasing (i.e. 0<=an) and approaches 0, then the following alternating series converges:
If the series converges, then the remainder R,sub>N = S-SN is bounded by |R N|<=aN+1. S is the exact sum of the infinite series and SN is the sum of the first N terms of the series.
Deleting the first N Terms
The following series either both converge or both diverge if N is a positive integer.
Geometric Series Convergence
With the geometric series, if r is between -1 and 1 then the series converges to 1⁄(1-r).
See this article for how to calculate the geometric series.
Integral Series Convergence Test
The following series either both converge or both diverge if, for all n>=1, f(n) = an and f is positive, continuous and decreasing. If the series does converge, then the remainder RN is bounded by
If p > 1, then the p-series converges
If 0 < p < 1 then the series diverges
The following rules apply if for all n, n≠0. L = lim (n—>∞)|an+1⁄an|.
If L<1, then the series converges.
If L>1, then the series diverges.
If L=1, then the ratio test is inconclusive.
Taylor Series Convergence
The Taylor series converges if f has derivatives of all orders on an interval “I” centered at c, if lim(n–>∞)RN=0 for all x in l:
The Taylor series remainder of RN=S-SN is equal to (1/(n+1)!)f(n+1)(z)(x-c)n+1 where z is a constant between x and c.
Word Problems: Example
Step 1: Read through the word problem carefully. Don’t skim or skip over phrases and sentences that may seem unimportant. You may miss details that change the entire meaning of the passage. Once you’ve read through the problem once, write down the answer that the question is asking for. Note the format that you need to give the answer in. For example, you might have to give certain units.
Step 2: Break the word problem down into individual statements about the scenario described in the word problem. Record any numbers or equations given in these statements. Note the units of measurement.
Step 3: Either extract the main equation from the word problem or build the equation using the information presented in the statements. Fill in as much of the equation as you can using the information from the word problem.
Step 4: Solve the equation to produce the answer needed to satisfy the word problem.
Step 5: Double-check your calculations and consider whether your answer makes sense in the context of the word problem.
- Be sure to state the answer in the format in which it was requested.
- If the equation remains unsolvable, you may need to extract or build more equations using the rest of the available but unused information to answer the empty spaces in your main equation. This is one of the more difficult parts of solving calculus word problems. It’s also one of the most important. Having a solid understanding of calculus, particularly the fact that derivatives represent the rate of change of the equation, will help you when creating the necessary equations.
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