CALC

Question 1

What are derivatives:

Answer 1

Derivatives are the rates at which quantities change. They measure how a function behaves as its input (x-value) changes.

The derivative is the slope of the tangent line

Also the instantaneous rate of change

Question 2

∆Y/∆X

Answer 2

Rate of change. ΔY/ΔX represents the average rate of change between two points on a curve or line. It measures how much y changes for every unit increase in x.

Question 3

Instantaneous Rate of Change

Answer 3

The instantaneous rate of change refers to the rate at which a function is changing at a specific point. It measures how quickly the output of a function is changing with respect to the input at that particular instant.

The derivative is the instantaneous rate of change

Question 4

Limits

Answer 4

Limits are used in calculus to describe the behavior of a function as it approaches a certain value or point. It helps determine what happens to the output of a function when the input gets closer and closer to a specific value.

Question 5

Approaching x

Answer 5

Approaching x means getting closer and closer to the value of x without actually reaching it.

Question 6

Denominator

Answer 6

The denominator is the bottom part of a fraction that represents the total number of equal parts into which a whole is divided.

Question 7

Indeterminate

Answer 7

0/0 - When plugging in a value to a limit gives as 0/0. Instead plug in values on left and right of x value

Can also use L’Hospitals

Question 8

Difference Rule

Answer 8

The difference rule is a derivative rule that allows us to find the derivative of a function by subtracting the derivatives of its individual terms.

Question 9

Root Rule

Answer 9

The root rule is a derivative rule used to find the derivative of a function that involves a radical expression. It states that the derivative of √(x) is equal to 1/(2√(x)).

Question 10

Sum Rule

Answer 10

The sum rule is a calculus rule that states that the derivative of the sum of two functions is equal to the sum of their derivatives.

Question 11

Algebraic limits

Answer 11

Algebraic limits are limits that involve algebraic expressions, where you need to evaluate the behavior of a function as it approaches a certain value or infinity.

Question 12

Rational Functions

Answer 12

Rational functions are functions that can be expressed as a ratio (fraction) of two polynomials. They have both a numerator and denominator, where neither can be zero simultaneously

Question 13

Approximation

Answer 13

An approximation is an estimate or close value that is not exact but provides a reasonable representation of something. It involves finding values that are close enough to be useful while acknowledging some degree of error.

Limits are approximations

Question 14

Asymptote

Answer 14

An asymptote is a line that a graph approaches but never touches. It can be horizontal, vertical, or slanted.

Question 15

Conjugates

Answer 15

Conjugates are pairs of complex numbers or algebraic expressions that have the same real part but opposite signs for their imaginary parts. In algebra, conjugates are often used in rationalizing denominators or simplifying radical expressions.

Question 16

Direct substitution

Answer 16

Direct substitution is an algebraic method used to evaluate limits by plugging in the desired value directly into the expression. It works when there are no undefined values or indeterminate forms involved.

Question 17

Integrals

Answer 17

Integrals are mathematical tools used to find the area under a curve or to calculate the accumulation of quantities over a given interval

Question 18

Polynomial

Answer 18

Polynomials are algebraic expressions consisting of variables, coefficients, and exponents, combined using addition, subtraction, multiplication, and non-negative integer exponents.

Question 19

Trigonometric Identities

Answer 19

Trigonometric identities are equations that relate different trigonometric functions. They allow us to simplify and manipulate trigonometric expressions.

Question 20

Jump Discontinuity

Answer 20

The limit from the right does not equal the limit from the left

The lines don’t match up

Question 21

Removable Discontinuity

Answer 21

A single dot away from the graph

A factored function

Limit at a point does not exist

Question 22

Asymptote Discontinuity

Answer 22

The limit goes to infinity

There are two lines that point straight up or down

Question 23

Closed Interval

Answer 23

A closed interval is a set of real numbers that includes both of its endpoints. It is denoted by square brackets [ ].

Question 24

Open Interval

Answer 24

An open interval is a set of real numbers between two endpoints, where the endpoints are not included in the interval.

Question 25

Horizontal Asymptote

Answer 25

This term refers to a horizontal line that a graph approaches but never crosses as x goes towards positive or negative infinity

Question 26

Unbounded Behavior

Answer 26

Unbounded behavior refers to the behavior of a function as the input approaches infinity or negative infinity. It describes how a function grows or decreases without any limit

Question 27

Tangent line

Answer 27

A tangent line is a straight line that touches a curve at only one point without crossing through it. In calculus, we use tangent lines to approximate curves and find instantaneous rates of change

Question 28

Composite Functions

Answer 28

Composite functions are formed by combining two or more functions, where the output of one function becomes the input of another. It's like putting one function inside another to create a new function.

Question 29

Differentiation

Answer 29

Differentiation is the process of finding the rate at which a function changes. It involves calculating the derivative of a function to determine its slope at any given point.

Question 30

Higher-Order Derivatives

Answer 30

Higher-order derivatives refer to taking derivatives multiple times. For example, if we take two derivatives of a function, we get its second derivative. Higher-order derivatives provide information about how fast rates change over time.

Question 31

Implicit Functions

Answer 31

Implicit functions are equations where the dependent variable is not explicitly expressed in terms of the independent variable(s). They often involve multiple variables and can be represented by curves or surfaces.

Question 32

Inverse Functions

Answer 32

Inverse functions are two functions that "undo" each other. When you apply one function and then the inverse function, you get back to where you started.

Question 33

Concave up

Answer 33

Concave up refers to the shape of a graph where it curves upward, resembling a smiley face. It indicates that the second derivative of a function is positive.

Question 34

Concave down

Answer 34

Concave down refers to the shape of a graph where it curves downward, resembling a frowning face. It indicates that the second derivative of a function is negative

Question 35

Mean Value Theorem

Answer 35

The Mean Value Theorem states that if a function is continuous on a closed interval [a, b] and differentiable on an open interval (a, b), then there exists at least one point c in (a, b) where the instantaneous rate of change (derivative) equals the average rate of change over the interval

Question 36

Secant Line

Answer 36

A secant line is a straight line that intersects a curve at two points. It represents the average rate of change between those two points on the curve.

Question 37

Absolute Extrema

Answer 37

The absolute extrema of a function are the highest and lowest values that the function reaches over a given interval.

Question 38

Global Extrema

Answer 38

Global extrema refer to both the maximum and minimum values of a function over its entire domain.

Question 39

Local Extrema

Answer 39

Local extrema are the highest or lowest points on a graph within a specific interval. They occur when the slope of the function changes from positive to negative (for a local maximum) or from negative to positive (for a local minimum).

Question 40

Relative Extrema

Answer 40

Relative extrema are points on a graph where there is either a local maximum (highest point) or a local minimum (lowest point). These points are compared to their neighboring points rather than the entire graph.

Question 41

Critical Points

Answer 41

points where the function’s derivative equals 0 0 or is undefined, and the points where the function itself is undefined. A function can change from increasing to decreasing here

Question 42

First Derivative Test

Answer 42

The First Derivative Test is a method used to determine the intervals on which a function is increasing or decreasing, and to identify local extrema (maximum or minimum) points.

Question 43

Local Maximum

Answer 43

A local maximum refers to the highest point of a function within a specific interval. It is higher than all nearby points but may not be higher than all other points on the entire function.

Question 44

Local Minimum

Answer 44

A local minimum refers to the lowest point of a function within a specific interval. It is lower than all nearby points but may not be lower than all other points on the entire function

Question 45

Second Derivative Test

Answer 45

The second derivative test is used to determine whether critical points correspond to local maxima, minima, or neither. It involves analyzing the concavity of a function at those critical points.

Question 46

Quadratic Formula

Question 47

d/dx (sec (x))

Answer 47

sec (x) tan (x)

Question 48

Inflection point

Answer 48

Where change in curvature occurs