APEX Precalculus: A Calculus Prep Text

This section focuses on factoring techniques for polynomials, particularly quadratic and cubic functions. It emphasizes the importance of factoring for solving equations and simplifying rational functions. The text notes that higher-degree polynomials can be difficult to factor and are less frequently needed in calculus. The concept of irreducible quadratics (those that cannot be factored into linear factors) is introduced, along with the use of the quadratic formula to identify them. The section highlights the inverse relationship between expanding and factoring, emphasizing the need for proficiency in both to successfully manipulate algebraic expressions. An example of expanding a squared binomial with multiple terms is included, demonstrating the application of distribution and the combining of like terms for simplification. Methods for factoring are discussed and presented as problem-solving puzzles, helping students develop understanding instead of simply rote memorization.

This section covers techniques for solving equations and inequalities. For equations, the text highlights the importance of isolating the variable, a skill applicable to various equation types, including linear, quadratic, and non-polynomial equations. The quadratic formula is emphasized as a tool for solving quadratic equations, building upon the earlier discussion of factoring. Higher-degree polynomial equation solving is briefly addressed, noting the existence of formulas for cubics and quartics but highlighting their complexity and infrequent use. For inequalities, the approach involves converting the inequality into an equality to find 'break points,' where the inequality might change from true to false. The importance of finding all break points to accurately determine the solution set is stressed. The section emphasizes the need to adapt solution methods based on the equation type (linear versus quadratic, for instance) and the need for strong algebraic skills to effectively solve a range of problems.

This section covers function domains, which are the sets of all possible input values for a function. The significance of function domains in real-world applications and mathematical modeling is highlighted. The section emphasizes the need to ensure that the model accurately reflects both real-world and mathematical constraints. Furthermore, it discusses effective graphing techniques, moving beyond simple point-plotting to encompass an understanding of function shapes and transformations. The text discusses how shifts, stretches, and reflections of basic functions can be used to quickly sketch related functions. The connection between a function and its graph is emphasized as crucial for analyzing function behavior and interpreting key features. Different types of functions (even and odd polynomials, exponential, logarithmic) are briefly introduced, each with distinct characteristics useful for quick graphing.

This section details the technique of completing the square, a useful algebraic method frequently applied in integral calculus. The method is presented as a problem-solving approach rather than a rigid algorithm to encourage deeper understanding and retention. It explains how to identify and manipulate quadratic expressions to rewrite them in a specific form that is often beneficial for further calculations. The process involves manipulating the quadratic term to create a perfect square trinomial, allowing for simplification and revealing important information about the quadratic function. The section extends the technique beyond quadratics with a leading coefficient of 1, showing how to handle cases with different leading coefficients by identifying three parameters (a, b, and c) and using them to complete the square. The method's usefulness in calculus is emphasized, especially in simplifying expressions and finding key features of functions.

This section provides a foundational understanding of linear functions, crucial for many calculus concepts. It introduces two common forms for representing linear functions: slope-intercept form and point-slope form. The text explains how to determine the slope and y-intercept from these forms, emphasizing that familiarity with both and the ability to switch between them is beneficial. The relevance of the point-slope form in calculus is highlighted, particularly when dealing with lines over small regions where the y-intercept might not be readily apparent. The concepts of parallel and perpendicular lines are also discussed. Parallel lines have the same slope, while perpendicular lines have slopes that are negative reciprocals of each other. The section emphasizes the frequent appearance of linear functions in calculus, from approximating more complex functions to analyzing rates of change using secant and tangent lines.

This section focuses on exponential functions, which are frequently encountered in calculus and differential equations. The text defines exponential functions of the form f(x) = bˣ, where 'b' is a positive real number not equal to 1. Key features of exponential functions, particularly where b > 1, are described, including the points (0,1) and (1,b) that they always pass through. The rapid growth for positive x-values and the horizontal asymptote at y=0 for negative x-values are highlighted. The section explains how these functions are used to model growth rates in various real-world situations, such as population growth or financial investments, emphasizing that a solid grasp of exponential functions is vital for many calculus applications. Graphing techniques, including plotting key points and using the general shape to guide the graph, are also discussed.

The section explores logarithmic functions, emphasizing their close relationship to exponential functions. Logarithmic functions are described as reflections of basic exponential functions across the line y = x. Key features of logarithmic functions of the form f(x) = logb(x) are described. These include the points (1,0) and (b,1) that they always contain. The section also addresses the behavior of these functions for very small positive x values, and as x becomes very large. The importance of recognizing that the domain of a logarithmic function is (0, ∞) and that there is a vertical asymptote at x=0 (the y-axis) is strongly emphasized. The section mentions that logarithmic functions appear in many measurement scales like the Richter scale for earthquakes or decibel scale for sound, illustrating their practical applications. The relationship between logarithmic and exponential functions is highlighted as a key aspect for understanding their behavior.

This section introduces trigonometric functions, emphasizing their importance in calculus and their use in modeling natural phenomena such as sound and light waves. The text highlights that trigonometric functions are frequently used in related rates problems to determine how quickly quantities, like angles, are changing. The section explains that mathematicians typically view the input of a trigonometric function as an angle, and it introduces radians as the preferred unit for measuring angles in calculus, contrasting them with degrees. The text notes the relationship between radians, arc length, and circumference, further emphasizing the preference for radians in mathematical contexts because of this relationship. It also explains that radians simplify derivative calculations in calculus (a topic covered later).

This section describes methods for evaluating trigonometric functions, using both the unit circle and right triangles. Using the unit circle, cosine is presented as the x-coordinate and sine as the y-coordinate. This section highlights how scaling a right triangle doesn't change the angle values but scales the side lengths proportionally, impacting how trigonometric functions are calculated from side lengths. It explains that to handle scaling, cosine and sine should be calculated as ratios of side lengths (adjacent/hypotenuse and opposite/hypotenuse, respectively). This approach ensures consistency regardless of the triangle's size. The importance of understanding these different approaches to evaluating trigonometric functions is emphasized because one method might be more useful than another depending on the situation. The section also introduces inverse trigonometric functions, particularly arcsin(x), emphasizing the need for these functions in situations where angle values need to be determined from known side lengths.

This section reinforces the importance of using radians instead of degrees in calculus. The text explains that radians directly relate angles to arc length, unlike degrees. For a full circle (360 degrees), the arc length is the circumference (2πr), providing a clear connection between angle measurement and physical distance. This direct relationship makes radians particularly useful in calculus for calculations involving arc lengths and other geometric properties. The section provides a brief explanation about why mathematicians prefer radians (connection to arc length and simplified calculus derivatives). While the document does not explicitly state the trigonometric identities, it lays the groundwork for their later introduction. This foundational understanding of radians will be crucial for future topics in calculus involving trigonometric functions.

This section introduces partial fraction decomposition, a crucial technique for simplifying complex fractions that often arise in calculus, particularly in integral calculus and when working with inverse Laplace transforms in differential equations. The process involves breaking down a single, complex fraction into a sum of simpler fractions. The denominators of these simpler fractions are factors of the original denominator. The text explains that for linear factors in the denominator, the numerator of the simpler fraction will be a constant, while for quadratic factors, the numerator will be a linear expression. The technique involves strategically decomposing the original fraction and then using algebraic manipulation to determine the numerators of the simpler fractions. The overall goal is to transform a complex fraction into a sum of simpler fractions that are easier to work with in subsequent calculations, making it a fundamental skill for advanced calculus problems.

This section addresses solving complex equations, a skill heavily utilized in various areas of calculus. It emphasizes that the first step in solving for a variable is to identify the type of expression or function involved, which guides the appropriate solution process. The text contrasts approaches used for linear expressions versus quadratic or trigonometric expressions. For quadratic equations, the text recommends using the quadratic formula, particularly when dealing with complicated expressions that are difficult to factor. The section explains that quadratic equations will typically have two solutions, which may be the same (repeated root), distinct, or complex conjugates (involving imaginary numbers). The text also briefly touches on the existence of formulas for cubic and quartic equations, but notes their complexity and infrequent use in practice. For equations of degree 5 or higher, it's stated that no general formulas exist, emphasizing the limitations of direct algebraic methods for higher-degree polynomial equations. The section concludes by briefly considering non-polynomial equations, indicating that the first step typically involves isolating the non-polynomial term before proceeding with the appropriate solution method, which would be explained further in later sections.

This section discusses methods for solving systems of equations, specifically focusing on techniques useful in calculus, especially for finding points of intersection between functions. The text highlights that the most commonly used methods are substitution and elimination. Substitution involves solving for one variable in one equation and substituting that expression into the other equation, then solving for the remaining variable. The text recommends starting with the simpler equation if there's a choice. Elimination involves multiplying equations by constants, then adding the equations together in a way that eliminates one variable. This process simplifies the system to a single equation with one variable, making it easier to solve. The method's advantages become more apparent when dealing with multiple linear equations (as explained in later sections). This section emphasizes the importance of strong algebra skills for efficiently solving systems of equations, a skill vital for finding intersection points in integral calculus, for example, when calculating the area enclosed between multiple functions.

This section contrasts a tedious, point-by-point approach to graphing with a more efficient method that leverages an understanding of function shapes and transformations. The text cautions against relying solely on plotting numerous points to connect the dots, emphasizing that this method is time-consuming and can be challenging for complex functions. Instead, it advocates for using an overview of common function shapes as a starting point. This section explains how understanding the general shapes of common functions and how they are affected by shifts, stretches, and reflections (vertical and horizontal) can allow for a quicker, more accurate sketching of graphs. The text encourages quick sketches to visualize transformations, recommending that students work through transformations step-by-step (shift first, then stretch/shrink) until they become more comfortable combining steps. The importance of accurately graphing functions for analyzing their behavior in calculus is highlighted.

This section expands on graphing techniques by focusing on specific function types. It discusses the characteristics of even polynomials, highlighting their symmetry around the y-axis and how the sign of the highest degree term determines the behavior of both 'tails' of the graph. Exponential functions are also addressed, describing key features such as the points (0,1) and (1,b) they always pass through, their rapid growth for positive x-values, and their horizontal asymptote at y=0. Similarly, logarithmic functions are examined, noting their relationship to exponential functions as reflections across the line y=x. Key features such as their points (1,0) and (b,1), domain restrictions (0,∞), and vertical asymptote at x=0 are highlighted. The ability to quickly sketch these types of functions by leveraging their known properties is emphasized for its importance in calculus analysis.

This section explains the process of completing the square for quadratic functions where the coefficient of the squared term is 1. The method is presented as a puzzle-solving approach, encouraging a conceptual understanding rather than rote memorization of an algorithm. The text emphasizes that the goal is to rewrite the quadratic expression in the form (x + a)² + b. The strategy is to identify the value of 'a' using the linear term of the quadratic, then to determine 'b' by matching the constant terms. The text carefully walks through the reasoning behind each step, explaining why certain manipulations are performed (e.g., adding and subtracting a² to maintain the equality). The overall aim is to rewrite the quadratic expression in a form more suitable for further calculations in calculus, often simplifying expressions and revealing key properties of the function.

This section extends the technique of completing the square to quadratics with a leading coefficient other than 1. It notes that in these cases, the quadratic expression is rewritten in the form c(x + a)² + b, introducing a third parameter, 'c,' which affects both the quadratic and linear terms. The strategy changes slightly; now 'c' is determined first from the quadratic term. The text emphasizes that a common solution technique in mathematics involves working with higher-power terms first before moving to lower-degree terms. The method is illustrated with an example where the input variable is 't' instead of 'x', emphasizing that the underlying approach remains consistent regardless of the variable used. The section reinforces that completing the square is a valuable algebraic technique with applications in integral calculus, transforming expressions into forms suitable for various integration techniques.

This section emphasizes the critical importance of consistent units in calculations within the context of real-world calculus problems. The text highlights that many calculus problems are based on real-world scenarios, and these problems will inherently involve units of measurement (length, time, volume, area). It stresses that the final answer must include the correct units. The text explains that if units are inconsistent (e.g., one side of a square measured in feet, the other in inches), conversion is necessary to ensure a meaningful answer. The section uses the example of calculating the area of a square, demonstrating that using consistent units (both feet or both inches) is essential for an accurate result. This section establishes a foundational understanding of the role of units in problem-solving for real-world applications in calculus.

Building on the importance of unit consistency, this section discusses how units can provide insights into the calculation process. It explains that units can be written in fraction form (e.g., miles/hour for speed), indicating the type of operation involved (division of distance by time in this case). The text notes that some problems might begin with units but result in a unitless answer. Trigonometric functions are provided as an example; since they represent ratios of side lengths within a right triangle, if both sides have the same units, those units cancel out in the final calculation, resulting in a unitless ratio. This section highlights the need for careful attention to units, not only ensuring consistency but also correctly interpreting unitless results within the context of the problem. The emphasis is on developing an understanding of how units are handled in diverse calculations within a calculus framework.