Calculus: Derivative & Limits

This section establishes the fundamental concept of a derivative as a measure of the instantaneous rate of change. It connects the derivative to the slope of a tangent line, illustrating how to find the derivative using limits. The text emphasizes the practical significance of derivatives in understanding how quantities change over time. The example of dropping an object from a window and tracking its height (h) over time (t) clearly demonstrates how derivatives are used to analyze dynamic systems. The section further develops the concept of derivatives by introducing various algebraic techniques for finding derivatives, including methods for handling functions raised to constant powers. The importance of understanding the limitations of derivatives, such as points where the derivative does not exist (corners and cusps), is also highlighted. The concept of a derivative is connected to the idea of velocity (rate of change of position) and average speed, showing how the limit process provides increasingly accurate approximations. Several examples illustrate calculating average speeds over smaller time intervals to approach the instantaneous velocity, foreshadowing the formal definition of the derivative.

This section moves beyond the theoretical definition of derivatives to demonstrate their use in solving real-world problems. A key application is shown through an example of calculating the distance to Seattle based on observations of distance markers at different times. This problem highlights the use of the equation of a line (y = mt + b), where the slope (m) represents the speed and the intercepts provide significant information about the journey. The practical interpretation of slope, y-intercept, and t-intercept are discussed in detail within the context of this example. The section also explores the use of derivatives in determining whether lines are parallel. Geometric problems involving triangles and finding the equations of their sides are introduced, solidifying the link between geometrical concepts and calculus. Another example concerns a progressive tax system in the fictional Kingdom of Xyg, demonstrating how a piecewise-defined function can model real-world scenarios and showing how derivatives apply to such functions. The practical implications of the King's proposed tax system are discussed, highlighting the significance of the y-intercept.

The section delves into more advanced concepts. The challenges of applying the chain rule to implicit functions, where it’s not possible to explicitly solve for one variable, are discussed. Implicit differentiation, a technique to find derivatives in such situations, is explained. The text assures the reader that applying the chain rule will always result in a solvable equation for the derivative, even in complex scenarios. Further illustrating the power of derivatives, the section showcases a problem demonstrating that two families of curves are orthogonal—meaning their tangent lines are perpendicular at the points where they intersect. The concept of inverse trigonometric functions and their applications are briefly introduced. The section then addresses finding local and global extrema of functions, and it includes Fermat’s Theorem, which states that a differentiable function’s local extrema occur only at points where the derivative is zero. The significance of critical values, where the derivative is either zero or undefined, is highlighted as a crucial step in identifying potential maximum or minimum points. Examples are used to show how analyzing the behavior of derivatives near critical values helps determine whether a critical point is a local maximum, local minimum, or neither. This analysis involves checking if the derivative changes sign around a critical value. The section concludes with a discussion of global maximum and minimum problems within a specified interval, showing how calculus can be used to find the absolute highest or lowest value that a function attains on a given interval.

This section introduces the fundamental theorem of calculus, a cornerstone of integral calculus, which establishes the relationship between differentiation and integration. It explains how finding an antiderivative—a function whose derivative is the given function—allows for the calculation of definite integrals by evaluating the antiderivative at the limits of integration and subtracting the results. The section acknowledges the difficulty in finding antiderivatives, noting that unlike differentiation, there's no straightforward set of rules for all functions. It highlights the importance of the fundamental theorem as a tool for solving problems that result in sums of a specific form. The challenge of finding antiderivatives is emphasized, indicating that while there are techniques to aid in the process, it's not always a mechanical procedure and may require creativity and strategic approaches to rewrite functions. The text hints at the usefulness of techniques like partial fraction decomposition for integrating rational functions, although it states that it will not delve into the details for high-degree polynomials.

The section delves into strategies for finding antiderivatives. It emphasizes a technique of making educated guesses for the 'inside function' when dealing with compositions of functions, followed by algebraic manipulation to determine the remaining parts of the function. This approach is proposed as a more systematic alternative to simply rewriting the function in a suitable form. The text acknowledges that this approach might need multiple attempts or might not always work, but presents it as a useful method for simplifying the process of finding antiderivatives. It mentions the existence of a general technique called 'partial fractions' for integrating rational functions, but notes the algebraic complexity involved when the denominator's degree is higher than 2, along with the fact that factoring the denominator isn't always feasible. The discussion is restricted to integrating rational functions where the denominator is a quadratic polynomial (ax² + bx + c), implying that more complex cases would be handled in separate, more advanced sections.

This section shifts from analytical integration methods to numerical techniques for approximating definite integrals. It begins by contrasting the simpler method of approximating areas with rectangles to a more accurate method using trapezoids. The text argues that trapezoids provide a considerably better approximation for the area under a curve over small intervals. The importance of error estimation in numerical methods is highlighted, emphasizing the need to know how accurate an approximation is. The section discusses the concept of error estimation, defining the error as the difference between the approximation and the true value and explaining the need for an error estimate (E) that’s guaranteed to be larger than the actual error. The objective is an error estimate E(∆x) that decreases rapidly as the interval width (∆x) decreases, facilitating improved accuracy through finer interval divisions. The text states that for many functions, a suitable error estimate exists for the trapezoid approximation, ensuring a reliable bound on the error. The section further alludes to a more advanced numerical integration technique using parabolas to approximate the curve across pairs of subintervals, which yields a simpler formula for the area under the parabola; however, detailed algebraic derivations are omitted.

The final part of this section focuses on the application of integration. It shows how to calculate the area between two curves by approximating the area using rectangles. The section illustrates how to handle situations where one curve dips below the x-axis, demonstrating that even with negative function values, the height of the rectangle is still the difference between the functions’ values at each point. The text provides an example of finding the area between two specific curves over a given interval. The connection between integrals and net distance is discussed. The explanation clarifies that the integral of a velocity function represents the net displacement (net distance traveled), while to find the total distance traveled, it’s crucial to consider intervals where velocity is positive or negative separately, summing the absolute values of the respective integrals. An example involving an object thrown upwards and its resulting velocity function showcases this concept, illustrating how to account for changes in the direction of motion when calculating total distance.

This section demonstrates the application of calculus to optimization problems. A key example involves finding the dimensions of a rectangle with a fixed area that minimizes its perimeter. The problem uses the concept of derivatives to find critical points, determining where the derivative is zero. The second derivative test is then employed to confirm whether a critical point represents a local minimum. In this case, the solution reveals that the rectangle with the smallest perimeter is a square. The importance of considering the domain of the problem (positive side lengths) is emphasized, highlighting the need to evaluate the function's behavior within the realistic constraints of the problem. Another example involves finding the quickest route from a point on land to a point in a wading pool, involving different running speeds on land and in water. The solution involves setting the derivative of the total time function (which is a function of the distance at which the person enters the water) equal to zero, resulting in Snell's Law, which also governs light refraction.

The section introduces the concept of related rates, focusing on scenarios where multiple variables change over time. It starts by asking what quantities' rates of change are known and which one needs to be determined. A problem involving a swing illustrates this; the horizontal movement of the swing's point of attachment is known (dx/dt), and the goal is to find the rate at which the swing is rising (dy/dt) or the rate of change of the angle (dθ/dt). The Pythagorean theorem is employed to relate the variables, and implicit differentiation is used to find the desired rates of change. The solution emphasizes the need to understand which variables are changing with time and which ones remain constant, and it uses a specific instance of a right triangle with sides 6-8-10 to solve the problem in terms of degrees per second. It also highlights that in some cases, what appears to involve multiple variables might reduce to just two, simplifying the process of taking derivatives. The section goes on to explain how to handle cases involving multiple variables that truly change over time. The approach involves taking the derivative of the equation relating all variables, substituting known values, and solving for the unknown rate. This approach is demonstrated using a problem involving a right triangle whose sides change simultaneously. This example showcases a problem where all three sides are variables, even if only a specific set of side lengths is of interest in the final answer.

This section presents a variety of additional applications, emphasizing the broad scope of calculus in various fields. Problems involving finding the fraction of a triangle's area occupied by the largest inscribed rectangle are discussed, demonstrating how to apply optimization techniques to geometric problems. Another problem explores how changes in production costs (due to economies of scale) affect the optimal number of items to produce. An example about finding the position of a particle between two electrical charges that minimizes the force on it shows that calculus can model physical interactions effectively. The problems are designed to showcase how calculus can be applied to diverse contexts, illustrating the power of calculus in modeling and solving real-world problems. The section contains various optimization problems, for example, determining what percentage of a sphere’s volume is occupied by the cone with the largest possible surface area. Another problem involves determining the location of a positive charge on the line between two opposite charges which minimizes the force pulling it towards the negative charge. These problems cover different areas of application, such as geometry, physics, and potentially economics.

This section provides a rigorous definition of limits, emphasizing the concept of a function's value approaching a specific number as the input approaches a particular value. It highlights that the formal definition involves ensuring that the function's output remains within a specified range around the limit value, regardless of whether the input approaches from the left or the right. This leads to the concept of a limit existing only if the function values are close to the limit from both sides. The text explains that the limit might not exist even if the one-sided limits exist. The section introduces the idea of one-sided limits, acknowledging that in some cases, the limit from only one side (left or right) might exist even if the overall limit doesn't exist. The importance of one-sided limits is underscored as a useful concept for discussing the behavior of functions at points where the overall limit is undefined, especially considering practical applications. The use of epsilon-delta notation is suggested but not explicitly detailed, as is common in introductory calculus texts that aim to build intuition before delving into a formal epsilon-delta proof.

This section focuses on implicit differentiation, a technique for finding derivatives of functions that are not explicitly defined. The method involves taking the derivative of both sides of an equation relating x and y with respect to x, treating y as a function of x and applying the chain rule. The text states that this always leads to an equation that can be solved for dy/dx, the derivative of y with respect to x. A simple example showcasing implicit differentiation is described. The authors emphasize the utility of implicit differentiation when solving for y directly is difficult or impossible. It's shown as a practical technique for finding the derivative without needing to explicitly solve the equation for one variable. The section also introduces the concept of families of curves and their orthogonality, relating it to implicit differentiation and using the method to solve problems concerning orthogonal families of curves, that is, families whose tangent lines are perpendicular at intersection points.