Calculus

In calculus, exponential functions with a base other than e (like ax) and logarithms with base a (like loga x) are handled by converting them into their “natural” forms. Since the natural exponential ex and natural log ln(x) have very simple derivatives, we use them as the bridge to understand ax and loga x. Interactive ... Read More

Resource: The Exponential Function Suppose we have a quantity \(y\), whose rate of change over time is proportional to the amount present. We can describe this relationship using a differential equation: \[ \frac{dy}{dt} = ky \] If we define the initial state where \(y = y_0\) at time \(t = 0\), the solution to this ... Read More

Resources: What is Natural Logarithm? In calculus, the natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.71828. Unlike base-10 logarithms (common logs), the natural logarithm, denoted as $ln(x)$, is preferred in calculus because it simplifies the differentiation and integration of exponential functions. 1. ... Read More

Resources: What is Euler’s Number? Euler’s number (e≈2.71828e is approximately equal to 2.71828) was first discovered by Swiss mathematician in Jacob Bernoulli in1683 while studying compound interest. However, it is named after Leonhard Euler, who popularized the constant, discovered its connection to calculus, calculated its value to 23 decimal places, and used the letter 𝑒 ... Read More

Resources: What is Inverse Function in Calculus? In calculus, an inverse function is a function that “undoes” the action of another function. If a function $f$ takes an input $x$ and gives an output y, its inverse $f^{−1}$ takes $y$ and brings you back to $x$. Mathematically: $f(x) = y \iff f^{-1}(y) = x$ 1. ... Read More

References: Integrals can be used to calculate many things such as area between curves, volumes and surfaces of solids, length of curves etc. Areas between Curves: In calculus, we define the exact area under a curve not by measuring it directly, but by slicing it into rectangles, adding them up, and then making the slices ... Read More

Book Calculus by Thomas/Finney, 9th Edition, Page 275 Differential Calculus to Integral Calculus. This is where we stop asking “how fast is it changing?” and start asking “how much have we accumulated?” Indefinite Integrals The chapter kicks off with a simple question: If I give you the answer (the derivative), can you tell me the ... Read More

Book Calculus by Thomas/Finney, 9th Edition, Page 248 Linearization: Approximating functions https://math.stackexchange.com/questions/1385936/approximating-sqrt1-frac1n-by-1-frac12n#:~:text=The%20function%20f(%20x)=%20%E2%88%9A%201+%20x,%E2%88%9A%201+%20x%20for%20x%20%E2%89%A5%200. Differentials Differential Estimate of Change:In calculus, estimating change with differentials is a method from calculus used to approximate how much a function’s output changes when the input changes by a small amount. Instead of calculating the exact change (which can be difficult and ... Read More

Book Calculus by Thomas/Finney, 9th Edition, Page 233 Example problem: Find 2 positive numbers whose sum is 20 and their product is as large as possible.