A horizontal line The chain rule tells us how to differentiate function compositions: True or False If the slope of a linear function is zero, the graph of the function is: A function is increasing... State the exponent rules and give an example of each.

If the slope of a linear function is zero, the graph of the function is:

The y-value decreases as the x-value increases. Which statement is true about the graph of y = -1/2x + 3? Describe the domain and range for both logarithmic and exponential functions. What is the shape of a quadratic function? Identify its distinguishing features. Concave up

Which statement is true about the graph of y = -1/2x + 3?

a) the zeros are where f(x) = 0 b) the zeros are the same as the x-intercepts c) you can find the zeros of a rational function by setting the numerator equal to 0 The derivative of x^n, where n is a constant, is n times x raised to the power n - 1: True or False When finding absolute extrema, why do we need to check the endpoints of the interval? Select ALL of the true statements about the zeros of a function f(x) What is the definition of a function?

Select ALL of the true statements about the zeros of a function f(x)

lim x => a+ f(x) = lim x=> a- f(x) = L What is the first thing you should try when solving a limit problem? Which of the following is true if limit x => a f(x) = L The second derivative represents the rate of change of the slope of the function: True or False The derivative of 1/x is x^-2: True or False

Which of the following is true if limit x => a f(x) = L

As x approaches a, f(x) approaches L The derivative of the position function with respect to time represents the acceleration of an object: True or False THe derivative of 1/x^3 is x^2: True or False What does it mean for lim x=> a+ f (x) = lim x=> a- f(x) = L How is the slope calculated?

What does it mean for lim x=> a+ f (x) = lim x=> a- f(x) = L

g (x) = 4 + x^4 - x^2 The chain rule tells us how to differentiate function compositions: True or False What is a tangent line? What is the shape of a quadratic function? Identify its distinguishing features. Polynomial function (4th degree) example

Polynomial function (4th degree) example

True Which statement is true about the graph of y = -1/2x + 3? Which of the following is true if limit x => a f(x) = L The derivative of a constant term is always zero: True or False What are the zeros of a function, f(x)? What is the difference between finding the zeros and evaluating the function at zero?

The derivative of a constant term is always zero: True or False

Calculus Final (Previous exam questions) Flashcards

A horizontal line

The y-value decreases as the x-value increases.

a) the zeros are where f(x) = 0 b) the zeros are the same as the x-intercepts c) you can find the zeros of a rational function by setting the numerator equal to 0

lim x => a+ f(x) = lim x=> a- f(x) = L

As x approaches a, f(x) approaches L

y = 7 - 2x

y = 6x + 9x^2 - 12

g (x) = 4 + x^4 - x^2

f (w) = 4^w

True

False

True

False

True

False

True

False

True

False

True

False

True

Second Derivative Test

The function f is concave up at a

To find max/min values that might occur there

They always occur at critical points

The intervals where a function is increasing or decreasing

a) There is a local maximum at x = c b) There is a vertical asymptote at x = c

A point on the function where the derivative is 0 or undefined - Where the derivative switches from positive to negative

Peak/valley - Always occur at critical points - Highest/lowest place in its neighborhood - 2nd derivative (critical points into f''(x)) - 1st derivative (number line and intervals)

Plug critical points and endpoints into ORIGINAL - Absolute maximum and minimum - x and y values

1) Find critical points 2) Number line to test intervals 3) State where f is increasing and decreasing

The tangent line has a positive slope, f'(x) > 0 - uphill

The tangent line has a negative slope, f'(x) < 0 - Downhill

Break up concavity intervals - f''(x) = 0

Derivative is increasing - f'(x) increasing - f'' > 0 is positive

Derivative is decreasing - f'(x) decreasing - f'' < 0 is negative

1) Find critical points 2) Plug into the second derivative f"(cp) > 0 => local minimum f"(cp) < 0 => local maximum

A function is a relation that assigns exactly one output (or value in the range) to each input (or value in the domain). In simpler terms: A function takes an input, performs a specific rule or operation, and gives back only one output for each input. Mathematically: If f is a function and x is an input in its domain, then: f(x)= output For every x, there is only one f(x)

The vertical line test is a visual way to determine whether a graph represents a function. Definition: The vertical line test states that: If any vertical line intersects the graph in more than one point, then the graph does not represent a function. What it determines: It tells you whether a graph passes the "one output for each input" rule — the key definition of a function. Why it works: A vertical line represents a single input (an xx-value). If that line touches the graph at more than one point, then the input is producing more than one output — which violates the definition of a function.

The domain of a function is the set of all possible input values (usually x-values) that the function can accept without causing any mathematical issues like division by zero or square roots of negative numbers.

The range of a function is the set of all possible output values (usually y-values or f(x) that the function can produce based on its domain.

The zeros of a function f(x) are the values of x where the function equals zero. In other words, the zeros are the x-coordinates where the graph of the function intersects the x-axis. Mathematically, the zeros are the solutions to the equation: f(x)=0 Evaluating the function at zero means finding the output of the function when the input x is zero. This is simply computing f(0) Finding the zeros of a function involves solving for the values of x where f(x)=0 Evaluating the function at zero means simply plugging x=0 into the function and calculating the result.

Set the function equal to zero: Write the equation f(x)=0f(x)=0, where f(x)f(x) is your given function. Solve the equation: Solve the equation for xx. This may involve different algebraic techniques, depending on the form of the function. If it's a polynomial, factor the expression or use the quadratic formula (for quadratic equations). If it's a rational function, set the numerator equal to zero (and exclude any values that make the denominator zero). If it's a radical function, isolate the radical and square both sides (while checking for extraneous solutions). If it's a trigonometric function, solve using known trigonometric identities or inverse functions.

Function composition is the process of combining two functions to create a new function. You do this by applying one function to the result of another. Mathematically, if you have two functions f(x) and g(x), the composition of f and g, written as (f∘g)(x), means you first apply g(x) and then apply f(x) to the result. Order matters: (f∘g)(x)≠(g∘f)(x), unless the functions are specially related. Domain restrictions: When composing functions, make sure the domain of the inner function g(x) matches the domain of the outer function f(x).

A linear function is a function that creates a straight line when graphed. It has the general form: f(x)=mx+b Where: m is the slope of the line (how steep the line is). b is the y-intercept (the point where the line crosses the y-axis). Key Characteristics of a Linear Function: Constant Rate of Change: The slope m represents a constant rate of change, meaning that for every unit increase in x, the value of f(x) changes by a fixed amount. Straight Line: The graph of a linear function is a straight line. This means the function has no curves or bends. No Exponents or Nonlinear Terms: Linear functions do not have exponents on the variable x (e.g., no x2, x3x, etc.), no square roots, or other nonlinear operations.

The slope of a line is a measure of how steep the line is. It tells you the rate of change between two points on the line. The slope is usually denoted as mm, and it is calculated using the formula: m=y2−y1x2−x1 Where: (x1,y1) and (x2,y2) are two points on the line. y2−y1 is the change in y (vertical change). x2−x1 is the change in x (horizontal change). Steps to Calculate the Slope: Identify two points on the line. You need the coordinates of two points, say (x1,y1) and (x2,y2). Subtract the y-values: Find the difference between the y-values of the two points: y2−y1. Subtract the x-values: Find the difference between the x-values of the two points: x2−x1. Divide the difference in y by the difference in x: m=y2−y1x2−x1 A positive slope means the line rises as you move from left to right. A negative slope means the line falls as you move from left to right. A zero slope means the line is horizontal (no rise or fall). An undefined slope (division by zero) means the line is vertical.

The y-intercept of a function is the point where the graph of the function crosses the y-axis. This occurs when the x-coordinate is zero. In other words, the y-intercept is the value of f(x) when x=0. Mathematically: The y-intercept is found by evaluating the function at x=0: f(0)=y-intercept How is it determined? For linear functions, the y-intercept is simply the constant term bb in the equation of the line f(x)=mx+b, where: m is the slope. b is the y-intercept. So, if the function is in the form f(x)=mx+b, the y-intercept is b. For other functions, the y-intercept is determined by setting x=0 in the equation and solving for f(0).

A quadratic function is a polynomial function of degree 2, meaning the highest exponent of the variable x is 2. The general form of a quadratic function is: f(x)=ax2+bx+c Where: a, b, and c are constants. a is the coefficient of x2 and determines the direction and width of the parabola. b is the coefficient of x and affects the position of the vertex horizontally. c is the constant term and represents the y-intercept, the point where the graph crosses the y-axis.

Key Features of a Quadratic Function: Shape: The graph of a quadratic function is a parabola. If a>0, the parabola opens upward. If a<0, the parabola opens downward. Vertex: The vertex is the highest or lowest point of the parabola, depending on the direction it opens. The x-coordinate of the vertex is found by: x=−b2a Axis of Symmetry: The axis of symmetry is the vertical line that passes through the vertex. Its equation is: x=−b2a Y-Intercept: The y-intercept is the value of the function when x=0, which is f(0)=c X-Intercepts (Roots): The x-intercepts (or roots) are the values of x where f(x)=0. These can be found using: Factoring (if possible). Quadratic formula:

The zeros of a quadratic function are the values of xx where the function equals zero, i.e., f(x)=0. These values represent the x-intercepts of the quadratic function, which are the points where the graph crosses the x-axis. For a quadratic function of the form: f(x)=ax2+bx+c To find the zeros, you need to solve the equation: ax2+bx+c=0a

To find the vertex of a quadratic function in the form y=ax2+bx+c, you can use the following formula for the x-coordinate of the vertex: x=−b/2a Once you have the x-coordinate, substitute it back into the quadratic equation to find the y-coordinate of the vertex. So, the steps are: Find the x-coordinate of the vertex using x=−b/2a Plug this value of x into the original quadratic equation to find the corresponding y-coordinate. The vertex is then the point (x,y).

1. Product of Powers Rule When multiplying two powers with the same base, add the exponents. am×an=am+n Example: x3×x2=x3+2=x5 2. Quotient of Powers Rule When dividing two powers with the same base, subtract the exponents. aman=am−na Example: y5y2=y5−2=y3 3. Power of a Power Rule When raising a power to another power, multiply the exponents. (am)n=am×n Example: (x2)3=x2×3=x6 4. Power of a Product Rule When raising a product to a power, raise each factor in the product to the power. (ab)m=am×bm Example: (2x)3=23×x3=8x3 5. Power of a Quotient Rule When raising a quotient to a power, raise both the numerator and the denominator to the power. (ab)m=ambm(ba)m=bmam Example: (3xy)2=32x2y2=9x2y2 6. Zero Exponent Rule Any non-zero number raised to the power of zero equals 1. a0=1(for a≠0) Example: 70=1 7. Negative Exponent Rule A negative exponent means to take the reciprocal of the base and then apply the positive exponent. a−m=1/am Example: x−3=1x3 8. Fractional Exponent Rule A fractional exponent represents both a root and a power. am/n=amn Example: x12=x

The most basic form of an exponential function is: f(x)=a⋅bx where: a is the initial value or vertical stretch/shrink factor. It represents the value of the function when x=0 (i.e., the y-intercept), and it determines how the graph is vertically stretched or compressed. If a is positive, the graph will rise; if negative, it will fall. b is the base of the exponential function. It controls the rate of growth or decay: If b>1, the function represents exponential growth (the graph increases as x increases). If 0

The graph of an exponential function has a characteristic curved shape that can either represent growth or decay depending on the base of the function. If the base b>1, the graph shows exponential growth, and it increases rapidly as x increases. The graph starts near zero on the left (as x→−∞) and rises steeply to the right. If the base 0

A logarithm is the inverse operation of exponentiation. It answers the question: "To what power must a certain base be raised to produce a given number?" The logarithmic function can be written as: log⁡b(y)=x This means: bx=y Where: b is the base of the logarithm, which is the number being raised to a power. x is the exponent (the logarithm). y is the result of raising the base to the power x.

1. Product Rule The logarithm of a product is the sum of the logarithms of the factors. log⁡b(xy)=log⁡b(x)+log⁡b(y) 2. Quotient Rule The logarithm of a quotient is the difference of the logarithms of the numerator and the denominator. log⁡b(xy)=log⁡b(x)−log⁡b(y) 3. Power Rule The logarithm of a number raised to a power is the exponent times the logarithm of the base. log⁡b(xn)=n⋅log⁡b(x) 4. Change of Base Formula You can change the base of a logarithm using the following formula: log⁡b(x)=log⁡c(x)log⁡c(b) where c is any positive number (usually base 10 or base e). 5. Logarithm of 1 The logarithm of 1 in any base is always 0. log⁡b(1)=0 for any base b 6. Logarithm of the Base The logarithm of a number to its own base is always 1. log⁡b(b)=1 7. Logarithm of a Negative Number The logarithm of a negative number is undefined in the set of real numbers because no real number exponent will yield a negative result. Logarithms are only defined for positive values of x. 8. Natural Logarithm (ln) The natural logarithm is the logarithm with base ee, where e≈2.718 ln⁡(x)=log⁡e(x)

The graph increases slowly and continuously as x increases. It is the inverse of an exponential function f(x)=bx, so its graph is a reflection of that across the line y=x Key Features: Domain: x>0 Logarithms are only defined for positive x-values—you cannot take the log of zero or a negative number in the real number system. Range: (−∞,∞) The output (y-values) can be any real number. Vertical Asymptote: x=0 As x approaches 0 from the right, the function drops toward negative infinity. The graph never touches or crosses the y-axis. Intercept:The graph passes through: (1,0) Because log⁡b(1)=0 for any base b. Behavior: For 01: the graph increases, but very slowly. Visual Summary: Steep drop to the left (as x→0+) Gentle upward slope to the right Never touches the y-axis Passes through (1,0)

???? Exponential Function: General form: f(x)=a⋅bx(where a≠0,b>0,b≠1) Domain: (−∞,∞) You can plug in any real number for xx. Range: (0,∞) The output is always positive (assuming a>0); the function never touches or goes below zero. ???? Logarithmic Function: General form: f(x)=log⁡b(x)(where b>0,b≠1) Domain: (0,∞) You can only take the log of positive numbers. Range: (−∞,∞) The output can be any real number. ???? Key Relationship: Exponential and logarithmic functions are inverses of each other, so their domains and ranges are swapped:

The rate of change describes how one quantity changes in relation to another. In math, it's often used to show how the output (y) changes as the input (x) increases. It is most commonly calculated as: Rate of Change=Change in yChange in x=ΔyΔx ???? Rate of Change in a Linear Function For a linear function (like y=mx+b): The rate of change is constant. It is the slope of the line: m. Every time x increases by 1, y changes by the same amount m. ???? Rate of Change in a Quadratic Function For a quadratic function (like y=ax2+bx+c): The rate of change is not constant. It changes as x changes—this is because the graph is curved (a parabola). However, the second differences (the change of the change) are constant. Both describe how a function's output changes relative to the input. Both can be analyzed using tables or graphs. Both are used to interpret real-world scenarios like speed or growth.

A secant line is a straight line that intersects a curve at two or more points. In the context of a graph of a function: The secant line between two points on the curve shows the average rate of change of the function over that interval. It essentially connects the two points on the curve and gives you a straight-line approximation of the curve's behavior between them. Formula for the Slope of a Secant Line: If the secant line connects points (x1,f(x1) and (x2,f(x2), its slope is: Slope of Secant Line=f(x2)−f(x1)x2−x1 This is the average rate of change of the function from x1 to x2. A secant line intersects a curve at two points and represents average rate of change. A tangent line touches the curve at one point only and represents the instantaneous rate of change at that point.

A tangent line is a straight line that touches a curve at exactly one point and has the same slope as the curve at that point. ???? Key Characteristics: It represents the instantaneous rate of change of the function at a specific point. It just "grazes" the curve — it doesn't cross it near the point of contact. The slope of the tangent line at a point is the same as the derivative of the function at that point.

The difference between average and instantaneous rate of change lies in how much of the function you're analyzing and how quickly it's changing: ???? Average Rate of Change Measures how much a function changes over an interval. It's the slope of the secant line connecting two points on the graph. Formula: Average Rate of Change=f(x2)−f(x1)x2−x1 ✅ Use it when you're interested in the overall change between two values of x. ???? Instantaneous Rate of Change Measures how fast a function is changing at a single point. It's the slope of the tangent line at that point. Mathematically, it's the derivative: Instantaneous Rate of Change at x=a is f′(a) ✅ Use it when you're interested in the exact rate at one specific value of x. Example:For f(x)=x2, the instantaneous rate of change at x=3 is: f′(x)=2x⇒f′(3)=6

A tangent line is the limit of secant lines as the two points on the secant line get infinitely close together. ???? In Terms of a Limit: Given a function f(x)f(x), the slope of the secant line between xx and x+hx+h is: Secant Slope=f(x+h)−f(x)h As h→0, the two points on the curve (at x and x+h) merge into one, and the secant line becomes the tangent line. This limit gives the instantaneous rate of change at x, also known as the derivative: Tangent Slope (Derivative)=lim⁡h→0f(x+h)−f(x)h ???? Visual Connection: Secant line: Connects two points on the curve. Tangent line: What you get when those two points collapse into one. So, the tangent line is the ultimate version of the secant line—when the "distance" between the two points is essentially zero.

A limit describes the value a function approaches as the input (x) gets close to a certain number. But sometimes, it matters which direction you're approaching from: ???? Left-Handed Limit (Approaching from the left): You're getting close to a number from values less than it. Notation: lim⁡x→a−f(x) This reads as: "the limit of f(x) as x approaches a from the left." ???? Right-Handed Limit (Approaching from the right): You're getting close to a number from values greater than it. Notation: lim⁡x→a+f(x) This reads as: "the limit of f(x) as x approaches a from the right." ???? Why This Matters: In piecewise functions or discontinuous graphs, the left-hand and right-hand limits might not match. For a limit to exist at a point a: lim⁡x→a−f(x)=lim⁡x→a+f(x) If they don't match, then: lim⁡x→af(x) does not exist.

The first thing you should try when solving a limit problem is to directly substitute the value of x (the point you're approaching) into the function f(x). This is often the simplest and quickest method.

The limit definition for the slope of a tangent line is based on the concept of the instantaneous rate of change of a function at a specific point, and it is derived from the slope of a secant line. This is fundamental in calculus, as it connects derivatives (slopes of tangent lines) with limits and secant lines. Limit Definition for the Slope of a Tangent Line: The slope of the tangent line at a point x=a on the curve f(x) is defined as the limit of the slope of secant lines as the two points on the curve approach each other. Mathematically, this is expressed as: mtangent=lim⁡h→0f(a+h)−f(a)h Here: a is the point on the curve where you're finding the tangent line. h is the distance between a and a second point a+h on the curve. The fraction f(a+h)−f(a)h represents the slope of the secant line through the points (a,f(a))and (a+h,f(a+h) Why This Works: The slope of the secant line between two points (a,f(a) and (a+h,f(a+h) gives an average rate of change between those two points. As h approaches zero, the second point (a+h,f(a+h) gets closer to the first point (a,f(a), and the secant line approaches the tangent line. The instantaneous rate of change at the point x=a is the slope of the tangent line, and this is given by the limit of the secant line slopes as h→0 Connection to Secant Lines: Secant lines represent the average rate of change between two points on the curve. The tangent line represents the instantaneous rate of change at a single point. The key connection is that as the two points on the secant line get closer and closer (i.e., as h→0), the secant line approaches the tangent line. This means the slope of the secant line at points near x=a becomes closer and closer to the slope of the tangent line at x=a

A derivative represents the rate of change or slope of the tangent line of a function at a particular point. It tells us how a function is changing at any given moment and is the fundamental concept in differential calculus. In simpler terms, the derivative of a function at a point describes how quickly the function's output is changing as its input changes near that point. Derivative Notation: There are several notations for the derivative of a function f(x), depending on the context: f′(x): The most common notation for the derivative of f(x). ddxf(x): A more formal notation indicating the derivative of f(x) with respect to x. dydx: Used when the function is expressed as y=f(x), where y is the dependent variable and x is the independent variable. f(n)(x): The n-th derivative of f(x) (used for higher-order derivatives). Limit Definition for Finding a Derivative: The derivative of a function f(x) at a point x=a is defined as the limit of the average rate of change (the slope of the secant line) between two points, as the second point approaches a. This is called the difference quotient. The limit definition of the derivative is: f′(a)=lim⁡h→0f(a+h)−f(a)h Where: f′(a) is the derivative of f(x) at x=a, hh represents the small change in x, f(a+h)−f(a) is the change in the function's output, and The fraction f(a+h)−f(a)h represents the average rate of change (slope of the secant line) between x=a and x=a+h. As h→0, the two points on the secant line get closer, and the slope approaches the instantaneous rate of change(the slope of the tangent line).

The Derivative at a Point: The derivative of a function at a point x=a gives the slope of the tangent line to the curve f(x) at that specific point. It tells you the instantaneous rate of change at that particular point. The derivative at a point is represented as: f′(a)=lim⁡h→0f(a+h)−f(a)h Here, f′(a) is the slope of the tangent line to the graph of f(x) at x=a. The Derivative Function (or Derivative of the Function):The derivative function is a new function, often denoted f′(x), which provides the slope of the tangent line at every point x on the curve. It describes how the slope of the curve changes as x changes, giving the rate of change of the function at all points in its domain. The derivative function is obtained by applying the limit definition of the derivative to the entire function f(x), and its formula is: f′(x)=lim⁡h→0f(x+h)−f(x)h Here, f′(x) is a function that gives the slope of the tangent line at any point x. Derivative at a Point: Represents the slope of the tangent line at a specific point x=a. It's a single number that gives the instantaneous rate of change at x=a. Derivative Function: Represents a new function that gives the slope of the tangent line at any point x. It's a function f′(x) that describes how the slope (rate of change) varies with x.

The function ex is special because its derivative is the same as the function itself

1/x

Calculus Final (Previous exam questions)

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