Calculus 1

Question 1

What is the relationship between continuity and differentiability?

Answer 1

Differentiability implies continuity, but NOT vice versa.

This DOES NOT mean that if f is continuous at x=c then f is differentiable at x=c.

Question 2

What are the three conditions under which a function cannot be differentiable?

Answer 2

Question 3

For derivatives, what is the Power Rule?

Answer 3

Question 4

For derivatives, what is the Constant Rule?

Answer 4

Question 5

For derivatives, what is the Constant Multiple Rule?

Answer 5

Question 6

For derivatives, what is the Sum Rule?

Answer 6

The Difference Rule works simliarly. The derivative of a difference is equal to the difference of the individual derivatives.

Question 7

What is the derivative of the Sine function?

Answer 7

Question 8

What is the derivative of the Cosine function?

Answer 8

Question 9

What is the derivative of the constant ‘e’ raised to the power of ‘x’ ?

Answer 9

Question 10

What is the derivative of the natural log of ‘x’ ?

Answer 10

Question 11

For derivatives, what is the Product Rule?

Answer 11

Question 12

For derivatives, what is the Quotient Rule?

Answer 12

Question 13

What is the derivative of the Tangent function?

Answer 13

Question 14

What is the derivative of the Cotangent function?

Answer 14

Question 15

What is the derivative of the Cosecant function?

Answer 15

Question 16

What is the derivative of the Secant function?

Answer 16

Question 17

In Calculus, what is the Chain Rule?

Answer 17

Question 18

What is the derivative of:

Answer 18

Question 19

What is the derivative of the Common Logarithm?

The Common Logarithm is a logarithm with a base of 10.

Answer 19

Question 20

What’s the relationship between the derivative of the inverse of a function and the original function?

Answer 20

Question 21

What is the derivative of the Inverse Sine function?

Answer 21

Question 22

What is the derivative of the Inverse Cosine function?

Answer 22

Question 23

What is the derivative of the Inverse Tangent function?

Answer 23

Question 24

What is the Squeeze Theorem?

Answer 24

Question 25

What is the Intermediate Value Theorem?

Answer 25

In simpler terms, if a continuous function goes from one y-value to another, it must take on every y-value in between at some point.

Question 26

What is a useful formula for finding limits of functions as their input approaches infinity? (Not L’Hôpital’s rule)

Answer 26

If possible, manipulate the function so that all of the terms containing variables are of the following form:

Question 27

What is the definition of continuity?

Answer 27

A function f(x) is continuous at a point x = c if the limit of f(x) as x approaches c exists, is equal to f(c), and f(c) is defined.

Question 28

What is the Sum Property of Limits?

Answer 28

## Footnote This property works similarly for subtraction (the Difference Property of Limits)

Question 29

What is the Product Property of Limits?

Answer 29

Question 30

What is the Quotient Property of Limits?

Answer 30

Question 31

What is the limit of a function multiplied by a constant?

Answer 31

Question 32

What is the limit of a composite function?

Answer 32

## Footnote The outer function must be continuous at the limit of the inner function

Question 33

L'Hôpital's Rule

Answer 33

L'Hôpital's rule is a powerful tool in calculus used to evaluate limits of indeterminate forms like 0/0 or ∞/∞. It states that if the limit of a quotient results in one of these indeterminate forms, you can find the limit by taking the derivatives of the numerator and denominator separately and then evaluating the limit of the resulting quotient. ## Footnote If, after computing L'Hôpital's rule you still receive indeterminate form, it is possible that using the rule again on the new expression will result in a legitmate answer. This can be chained several times.

Question 34

Mean Value Theorem

Answer 34

Question 35

Definition of a Definite Integral

Answer 35

Question 36

Fundamental Theorem of Calculus

Answer 36

The Fundamental Theorem of Calculus (FTC) shows that differentiation and integration are inverse operations, meaning each operation undoes the other. There are two parts: the first part explains that you can find an antiderivative by integrating a function, and the second part shows you can calculate a definite integral by finding the difference between the values of its antiderivative at the endpoints of the interval.