∫ calculus

1. **State the problem:** We are given the implicit equation $$4x^2 + y^2 = 4$$ and need to find $$\frac{dy}{dx}$$ using implicit differentiation. 2. **Differentiate both sides wit

1. **State the problem:** We are given the implicit equation $$\sqrt[3]{x} + \sqrt[3]{y} = 8$$ and need to find $$\frac{dy}{dx}$$ using implicit differentiation. 2. **Rewrite the e

1. **State the problem:** Given the equation $$x^3 y^3 = 44$$, find $$\frac{dy}{dx}$$ using implicit differentiation. 2. **Differentiate both sides with respect to $$x$$:**

1. **State the problem:** We need to find $\frac{dy}{dx}$ by implicit differentiation for the equation $$\ln(2xy) = e^{xy}, \quad y \neq 0.$$\n\n2. **Differentiate both sides with

1. **State the problem:** We want to find $\frac{dy}{dx}$ by implicit differentiation from the equation $$\ln(2xy) = e^{xy}$$ where $y \neq 0$. 2. **Rewrite the equation:** $$\ln(2

1. We are asked to find the integral of $\sqrt{1+u^2}$. 2. Recall the integral formula for $\int \sqrt{a^2 + x^2} \, dx = \frac{x}{2} \sqrt{a^2 + x^2} + \frac{a^2}{2} \ln\left|x +

1. The problem is to find the integral of the function $1 + u^2$ with respect to $u$. 2. Recall that the integral of a sum is the sum of the integrals, so we can write:

1. The problem is to evaluate the integral $$\int \sqrt{1+u^2} \, du$$. 2. To solve this, we use a trigonometric substitution. Let $$u = \sinh(t)$$, so that $$du = \cosh(t) \, dt$$

1. **State the problem:** We need to determine whether the series $$\sum_{n=1}^\infty \left(\sqrt[3]{n^2 + 1} - n\right)$$ converges or diverges. 2. **Analyze the general term:** C

1. The problem is to analyze the given series and understand its behavior and graph. 2. The series is:

1. Problem: Calculate the limit $$\lim_{n \to \infty} \frac{n^4 + 5n^2 + n + 2}{3 + 4n - 5n^3 - 10n^4}$$ and similar rational functions. Step 1: Identify the highest power of $n$ i

1. Problem: Calculate the limit $$\lim_{n \to \infty} \frac{n^4 + 5n^2 + n + 2}{3 + 4n - 5n^3 - 10n^4}$$ and similar rational functions of polynomials. Step 1: Identify the highest

1. **State the problem:** We want to determine the convergence of the infinite series $$\frac{x}{1 \cdot 2} + \frac{x^2}{3 \cdot 4} + \frac{x^3}{5 \cdot 6} + \cdots$$

1. The problem is to find the integral of $\cosh\left(\frac{x}{a}\right)$ with respect to $x$. 2. Recall that the integral of $\cosh(u)$ with respect to $u$ is $\sinh(u) + C$.

1. Problem: Evaluate the definite integral $$\int_{0}^{1} 5 e^{2x-1} dx$$. 2. Step 1: Factor constants and simplify the integrand by writing $5 e^{2x-1}=5 e^{-1} e^{2x}$.

1. **State the problem:** Evaluate the definite integral $$\int_0^1 \frac{5}{e^{2x-1}} \, dx$$. 2. **Rewrite the integrand:** Since $$\frac{5}{e^{2x-1}} = 5 e^{-(2x-1)} = 5 e^{-2x+

1. **State the problem:** Find the limit $$\lim_{x \to 3} \frac{2x - 6}{\sqrt{x+1} - 2}$$. 2. **Identify the indeterminate form:** Substitute $x=3$ directly:

1. The problem involves using differentiation to find gradients, tangents, normals, stationary points (excluding points of inflexion), connected rates of change, small increments,

1. The problem is to know and use the derivatives of standard functions: $x^n$ (for any rational $n$), $\sin x$, $\cos x$, $\tan x$, $e^x$, and $\ln x$, with examples. 2. Derivativ

1. **Stating the problem:** Find the equation of the tangent line to a curve at a given point. 2. **Understanding the tangent line:** The tangent line to a curve at a point touches

1. The problem is to find the equation of the tangent line to a curve at a given point. 2. Suppose the curve is given by a function $y=f(x)$ and we want the tangent line at $x=a$.