∫ calculus

1. The problem is to understand what decreasing and increasing values mean in mathematics. 2. A function is called increasing on an interval if for any two numbers $x_1$ and $x_2$

1. **State the problem:** Find the limit as $x$ approaches 7 of the function $$\frac{5x^2 - 7x + 2}{x^2 - 1}$$. 2. **Recall the limit rule:** If the function is continuous at $x=7$

1. **State the problem:** Find $\frac{dy}{dx}$ by implicit differentiation for the equation $$2x^2 + xy - y^2 = 2.$$\n\n2. **Recall the rules:** When differentiating implicitly, tr

1. **State the problem:** We need to find $\frac{dy}{dx}$ by implicit differentiation for the equation $$2x^3 + xy - y^2 = 2.$$\n\n2. **Recall the formula and rules:** When differe

1. **Problem Statement:** We are given the function $$f(x) = x^4 + \frac{1}{2}x^3 - 5x^2 + \tan\left(\frac{x}{2}\right)$$ and asked to find at which of the given points $$x = -2, -

1. **Problem Statement:** We are given a function $f$ defined on the interval $[a,b]$ with $f(b) > f(a)$ and $a \leq x \leq b$. The derivative $f'(x)$ exists for all $x$ in $(a,b)$

1. **Problem Statement:** We are given a function $f$ defined on the interval $[a,b]$ with $f(b) > f(a)$ and the derivative $f'(x)$ exists for all $x$ in $(a,b)$ except at $x=0$. W

1. **Problem statement:** Calculate $\frac{dz}{dt}$ for the function $$z = f(x,y) = \sqrt{x^2 - y^2}$$ where $$x = e^{2t}$$ and $$y = e^{-t}$$. 2. **Formula and rules:** To find $\

1. **State the problem:** Find the volume of the solid obtained by rotating the region bounded by the curves $y = x^2 + 8x + 7$ and $y = 0$ about the x-axis. 2. **Identify the regi

1. **Problem Statement:** Expand $\log_e x$ in powers of $(x-1)$ and hence evaluate $\log_e 1.1$ correct up to 4 decimals. 2. **Formula and Explanation:** The Taylor series expansi

1. **State the problem:** Find the derivative of the function $$f(x) = \frac{\cot x}{1 + \csc x}$$. 2. **Recall formulas and rules:**

1. The problem is to find the derivative of a function, but the function is not specified in the question. 2. To find the derivative of a function $f(x)$, we use the definition of

1. **Problem Statement:** Find the Taylor series expansions of the following functions at the given points $a$. 2. **Recall:** The Taylor series of a function $f(x)$ at $x=a$ is gi

1. Let's start by stating the problem: We want to understand Taylor and Maclaurin series and how to analyze the pattern of derivatives $f^{(n)}(x)$ to find a general formula for th

1. **Problem statement:** Find the integral $$\int \frac{1}{(7x+4)^m} \, dx$$ for the cases (a) $$m=2$$ and (b) $$m=1$$. 2. **Formula and rules:** For integrals of the form $$\int

1. **Problem Statement:** Determine the Taylor series for the function $f(x) = x^3$ at $a=2$. 2. **Taylor Series Formula:** The Taylor series of a function $f(x)$ at $x=a$ is given

1. **State the problem:** We need to evaluate the definite integral $$\int_{0}^{4} \frac{10}{5x + 2} \, dx$$. 2. **Recall the formula:** The integral of $$\frac{1}{ax + b}$$ with r

1. **Problem statement:** Find the limit $$\lim_{x \to 2} \frac{\sqrt{x^2 + x} - \sqrt{6}}{2x - 4}$$. 2. **Recall the formula and rules:** When direct substitution results in an in

1. Let's start by stating the problem: We want to understand what Taylor and Maclaurin series are and how they are used to approximate functions. 2. A Taylor series of a function $

1. **State the problem:** We are given the function $$f(x) = \frac{x^3}{3} + \frac{3}{x^3} + 3\sqrt{x^5}$$ and want to simplify it and find its derivative. 2. **Rewrite the functio

1. **State the problem:** Evaluate the integral $$\int \sin(nx) + 2^x \cos x \, dx$$ where $n$ is a constant. 2. **Break the integral into two parts:**