Calculus Cheatsheet

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Cheatsheet

CALCULUS

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DERIVATIVES

INTEGRALS

LIMITS

LIMITS

A limit is defined as a value that a function approaches on the y-axis as the input approaches another value on the x-axis Limits are important in calculus and mathematical analysis and used to define integrals, derivatives, and continuity.

TYPES OF Limits

Algebraic

Limits at infinity

Graphical

SOLVING LIMITS GRAPHiCALLY

Like previously explained, L is the limit of f(x) as x approaches a, provided that as we get sufficiently close to a, from both sides without actually equaling a, we can make f(x) as close to L. As we approach a specific (finite) value along the x-axis from both the left and the right sides, we want to find what the y-value (L) is approaching.

SOLVING LIMITS GRAPHiCALLY

Using this logic, if we approach a limit from both sides, we could end up with different values depending on the side we chose to approach. If the two values obtained from approaching from both sides are not equal, it is said that the limit does not exist.

Continuity

If its possible to draw a limit function without lifting a pencil from the paper, we say that it is continuous. If not, the function has a discontinuity. The main types of discontinuity are removable, infinite and jump.

What is the value of the limit as it approaches -4?

EXERCISE

Zero

Does not exist

ANSWER

WRONG. TRY AGAIN

Next question

ANSWER

CORRECT. GOOD JOB

SOLVING LIMITS ALGEBRAICALLY

If you need to find the limit of a function algebraically, the two main techniques to choose are plugging in the x value and factoring. You can only use the first technique if the function is continuous at the x value at which you are taking the limit. If the function is undefined at this x value, you must move on to the second technique to simplify your function so that you can plug in the approached value for x.

For the first example, plugging in the value of the limit on the function we get an exact value of 3, this is the limit because the function is continuous at x=5. On the second example, plugging the value of the limit will result in a division of zero over zero, so a different approach must be taken. This is solved by factoring the quadratic in the numerator, which ends up as (x-4)(x-2). The two (x-4) cancel eachother and only f(x)= x-2 remains. If we plug x in this we get 2.

Examples

What is the value of the limit as it approaches 1?

EXERCISE

ONE THIRD

ONE FIFTH

Return to main menu

ANSWER

CORRECT. GOOD JOB

Return to main menu

ANSWER

WRONG. TRY AGAIN

Limits at INFINITY

Infinity is used to explain the end behavior of a function that is either increasing or decreasing without bound. For functions such as y=1/x, as x gets bigger and bigger, the y-values are getting smaller and smaller. This is the secret when limits approach infinity

In this example, we only substitute the value of infinity in the function and since dividing any number over infinity is close to zero, the value of the limit is determined by the independent term.

Example

What is the value of the limit as it approaches infinity?

EXERCISE

FIVE THIRDS

TWO THIRDS

Next Subject

ANSWER

CORRECT. GOOD JOB

Return to main menu

ANSWER

WRONG. TRY AGAIN

The derivative of a function describes the function's instantaneous rate of change at a certain point. Another common interpretation is that the derivative gives us the slope of the line tangent to the function's graph at that point.

DERIVATIVE

DERIVATIVE RULES

All constants have an imaginary x to the zeroth power, and using the constant power rule, we lower that exponent, multiply it by the constant and substract one from the original exponent. 0 times 4 is still 0, and every number or variable multiplied by 0 is 0, which is why all derivatives of constants are 0.

Constant rule:

EXAMPLES

In this case, the constant is taken out of the derivative first. The exponent of the variable is lowered as a multiple and one is substracted from the exponent. Finally we just multiply the exponent that was lowered and the original coefficient that was taken out, giving us our final answer.

Constant multiple rule:

EXAMPLES

First, we apply the derivative to both terms, using the constant power rule in each one of them. Both terms coefficient's are taken out of the derivative and one is substracted from the exponents of the variables. Finally, we just multiply the "new" coefficients by the originals that were taken out and get the final answer.

Sum rule:

EXAMPLES

Once again, we apply the derivative to both terms. Using the constant power rule in each one of them, both terms coefficient's are taken out of the derivative and one is substracted from the exponents of the variables. In the case of x, it is raised to 1, and 1-1=0, every number or variable raised to the zeroth is one. Finally, we just multiply the "new" coefficients by the originals that were taken out and get the final answer.

Difference rule:

EXAMPLES

The rule states that we take the derivative of the first term and multiply it by the second term, we then add to that the first term and multiply it by the derivative of the second term. In this case the derivative of the first term is 2x and the derivative of the second is simply 1. We multiply the derivatives with the original terms and get a polynomial expression that is simplified by grouping common terms.

Product rule:

EXAMPLES

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We have a complex function that we can call u raised to the tenth power. The chain rules that the derivative of f(x)=u^n is n times u^n-1 times the derivative of u. In this case the derivative of u is labeled in blue, so we just multiply by n, which in this case is 10, and u^n-1, which is inside the red parenthesis. We simplify the terms by common factor to get our final answer.

Chain rule:

EXAMPLES

The integral, also known as the antiderivative, represents in simple terms the area between the graph of a given function and the x-axis, specifically between two vertical lines, usually represented as a and b

INTEGRALS

The basic rules of integration state that we have to add one to the exponent of the variable, and that new exponent must be put under the variable as if it were a fraction. We can also take out constants out of integrals and use them after we're done with integration. In this case the integrated variable has a number 3 in the denominator that will cancel with the three that was multiplying the integral. Finally since this is in an indefinite integral, we have to add an integration constant represented as C.

EXAMPLES

Integration by substitution:

EXAMPLES

Integration by substitution:

EXAMPLES

Integration of e^u:

EXAMPLES

Integration of e^u:

EXAMPLES

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For definite integrals, we follow the same rules as if it were an indefinite integral, but instead of adding the integration constant at the end, we evaluate for the limits needed. We substitute in the integral and we susbtract the lower value from the higher. In this case since the substitution of the lower value gave us a negative number, this sign is going to multiply with the other negative sign, which will make that the two values will add in the end.

Definite integral:

EXAMPLES

THAT'S ALL FOLKS!

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