∫ calculus

1. **Problem statement:** Find the $n^{th}$ derivative of the function $f(x) = \tan(2x)$.\n\n2. **Recall the function and derivative rules:** The function is $f(x) = \tan(2x)$. The

1. **State the problems:** We need to evaluate the following limits using a table of values:

1. **State the problem:** Calculate the triple integral $$\int_0^3 \int_0^2 \int_0^{3-2x} dz\,dy\,dx$$.

1. **State the problem:** We want to evaluate the double integral $$\iint_D (2y + 4x) \, dA$$ where the region $D$ is bounded by the parabolas $y = 2x^2$ (lower boundary) and $y =

1. **Stating the problem:** Evaluate the double integral $$\int_0^1 \int_y^1 xy \cdot 8xy \, dx \, dy$$. 2. **Rewrite the integrand:** The integrand is $$xy \cdot 8xy = 8x^2 y^2$$.

1. **State the problem:** Evaluate the improper integral $$\int_0^{+\infty} 2 \ln(x) \, dx$$ and determine if it converges or diverges. 2. **Recall the integral and convergence rul

1. **State the problem:** Calculate the double integral $$\int_0^1 \int_0^y x y \cdot 8 x y \, dy \, dx$$. 2. **Simplify the integrand:** The integrand is $$x y \cdot 8 x y = 8 x^2

1. **State the problem:** Evaluate the improper integral $$\int_0^\infty 2 \ln(x) \, dx$$. 2. **Analyze the integral:** The integral involves the natural logarithm function multipl

1. **State the problem:** Determine if the limit $$\lim_{(x,y) \to (0,0)} \frac{x^2 - 3x}{4x - xy}$$ exists. 2. **Recall the rule:** For a two-variable limit to exist, the limit mu

1. Problem: Identify and explain the different types of discontinuity and give clear examples. 2. Key formula and rule: A function $f$ is continuous at $a$ iff $\lim_{x\to a} f(x)=

1. **Problem Statement:** Determine if a function is continuous on a given interval. 2. **Key Concept:** A function $f(x)$ is continuous on an interval if it is continuous at every

1. **Stating the problem:** Find the limit $$\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$$. 2. **Formula and rules:** The limit of a function as $x$ approaches a value $a$ is the value th

1. **Problem Statement:** Determine if the function $f(x)$ is continuous at a given point $x=a$. 2. **Definition of Continuity at a Point:** A function $f(x)$ is continuous at $x=a

1. The problem is to understand what special limits are in calculus. 2. Special limits are particular limits that frequently appear and have well-known results, which help in solvi

1. Let's start by stating the problem: We want to find the limits of some exponential, logarithmic, and trigonometric functions as the variable approaches a certain value. 2. For e

1. **State the problem:** Find the derivative of the function $f(x) = \sqrt{x} + 1$ using the first principle of derivatives (definition of derivative). 2. **Recall the definition

1. Let's start by stating the problem: We want to understand the concept of differentiation in IBDP Maths AI SL calculus. 2. Differentiation is the process of finding the derivativ

1. The problem is to find the derivative of the function $f(x) = \ln(3x^5)$. 2. Recall the derivative rule for the natural logarithm: if $f(x) = \ln(g(x))$, then $f'(x) = \frac{g'(

1. The problem asks to find the derivative with respect to $x$ of the expression $x^{1+x}$.\n\n2. The function is $y = x^{1+x}$. To differentiate this, we use logarithmic different

1. **State the problem:** Differentiate the function $$P(x) = 18 - \frac{x^2}{x+3} - \sqrt{x+3}.$$\n\n2. **Rewrite the function for clarity:** $$P(x) = 18 - \frac{x^2}{x+3} - (x+3)

1. **State the problem:** Find the derivative $P'(x)$ of the function $P(x) = x^3 e^x$. 2. **Recall the formula:** To differentiate a product of two functions, use the product rule