∫ calculus

1. **State the problem:** Evaluate the double integral $$\int_0^1 \int_x^1 y^2 e^{x^2} \, dy \, dx$$ by integrating first with respect to $y$ and then with respect to $x$. 2. **Int

1. **State the problem:** Solve the expression $x^3 - 5$ using first principles. 2. **Understand first principles approach:** This typically means evaluating or simplifying directl

1. **Stating the problem:** Use first principles (definition of a derivative) to find the derivative of the function $f(x) = x^3 - 5$. 2. **Recall the definition of the derivative:

1. Problem: Find the derivative of $$y = (2x^2 - 4x^3)^4$$. Step 1: Use the chain rule. Let $$u = 2x^2 - 4x^3$$, so $$y = u^4$$.

1. (b) Test the differentiability of $$f(x) = \begin{cases} x^{2} \cos\left(\frac{1}{x}\right), & x \neq 0 \\ 0, & x = 0 \end{cases}$$ at $$x=0$$. Step 1: Check continuity at $$x=0

1. Problem (b): Test the differentiability of $$f(x) = \begin{cases} x^2 \cos(\frac{1}{x}) & x \neq 0 \\ 0 & x=0 \end{cases}$$ at $$x=0$$. Step 1: Check continuity at $$x=0$$.

1. (b) Test differentiability of \[ f(x) = \begin{cases} x^2 \cos\left(\frac{1}{x}\right), & x \neq 0 \\ 0, & x = 0 \end{cases} \text{ at } x=0. \] Step 1. Check continuity at 0:

1. **Differentiate each given function:** 1. For $y = e^{x+2}$, use chain rule: $$\frac{dy}{dx} = e^{x+2} \times \frac{d}{dx}(x+2) = e^{x+2} \times 1 = e^{x+2}.$$

1. Let's start by defining a continuous function. A function $f(x)$ is continuous at a point $x = a$ if all the following conditions are met: - $f(a)$ is defined.

1. **State the problem:** We are given the function $$f(x) = 1 - 2kx + kx^2 - x^3$$ and asked to find the range of values for the parameter $$k$$ such that $$f(x)$$ is decreasing f

1. Let's first state the problem clearly: You want to know if a given series is a Maclaurin series or a Taylor series. 2. A Taylor series is an expansion of a function $f(x)$ aroun

1. We want to find the range of $x$ values for which the error in approximating $\sin x$ by $x - \frac{x^3}{6}$ is at most $5 \times 10^{-4}$. 2. The Taylor series of $\sin x$ abou

1. We are given that $f$ is continuous on $[a,b]$ and for every $x_1, x_2 \in [a,b]$, the inequality $f'(x_2) - f'(x_1) > 0$ holds when $x_2 > x_1$. This means the derivative $f'$

1. The problem states that for the function $g(x)$, we have $g'(2) = 0$ and $g''(2) = 5$. 2. To classify whether $x = 2$ is a local minimum, maximum, or an inflection point, we use

1. The problem is to find the x-coordinates of the inflection points of the function $f(x) = \sin(x)$ on the interval $[0, 2\pi]$. 2. Inflection points occur where the second deriv

1. We want to find where the function \( k(x) = \frac{1}{x^2 + 3} \) is convex downward. \n2. Recall that a function is convex downward where its second derivative \( k''(x) < 0 \)

1. We need to find the integral of the function $2x - 6$ with respect to $x$. 2. The integral of $2x$ is calculated as $$\int 2x \, dx = 2 \cdot \frac{x^2}{2} = x^2.$$

1. Stating the problem: Find the value of $$16(a^2 + b^2 + c^2)$$ given that

1. We are given the function $$f(x) = x + \frac{x^2}{2} + \frac{x^3}{3} + \frac{x^4}{4} + \frac{x^5}{5}$$ and its inverse function $$g(x) = f^{-1}(x)$$. We want to find the value o

1. **State the problem:** We want to find the limit as $x$ approaches 1 of the expression $$\frac{\sqrt{x+3} - 2}{3 - \sqrt{x+8}}$$

1. Stating the problem: Evaluate the integral $$\int \frac{3\sqrt{x} - 1}{5x - 6} \, dx.$$\n\n2. Rewrite the integral by expressing \(\sqrt{x}\) as \(x^{1/2}\):\n$$\int \frac{3x^{1