∫ calculus

1. The problem is to differentiate the function given by $$dy/dn = -\left(4y(0.5mmy + \sin \cos n \sqrt{1 + y} + 4 \sqrt{1 + n})\right)$$ with respect to $n$. 2. To differentiate t

1. The problem asks to find the limit $$\lim_{x \to 3} \frac{|x - 3|}{x - 3}$$. 2. This is a classic limit involving absolute value and a linear expression. The key is to consider

1. The problem asks to determine the x-values where the function $f$ is discontinuous and to specify if $f$ is continuous from the right, from the left, or neither at those points.

1. **Stating the problem:** Simplify the expression $2x'$ where $x'$ denotes the derivative of $x$ with respect to some variable (usually time or another independent variable). 2.

1. **Problem statement:** Evaluate the integral $$\int \frac{x \, dx}{(3 - 2x - x^2)^{3/2}}.$$\n\n2. **Rewrite the denominator:** The expression inside the power is $$3 - 2x - x^2.

1. We are asked to evaluate the integral $$\int \frac{x \, dx}{(3 - 2x - x^2)^{3/2}}.$$\n\n2. First, rewrite the quadratic expression in the denominator to a more recognizable form

1. **Problem (a):** Find the gradient of the curve $y = 3x^4 - 2x^2 + 5x - 2$ at points $(0, -2)$ and $(1, 4)$. 2. **Formula:** The gradient of a curve at any point is given by the

1. **Problem (a):** Find the gradient of the curve $y = 3x^4 - 2x^2 + 5x - 2$ at points $(0, -2)$ and $(1, 4)$. The gradient of a curve at a point is given by the derivative $\frac

1. **Problem statement:** Evaluate the integral $$\int t \sqrt{\frac{2+t^2}{2-t^2}} \, dt$$. 2. **Step 1: Simplify the integrand.** Write the integrand as $$t \sqrt{\frac{2+t^2}{2-

1. **Problem (a): Find the area enclosed by the curve $y = 4\cos 3x$, the x-axis, and the lines $x=0$ and $x=\frac{\pi}{6}$.** The area under a curve from $x=a$ to $x=b$ is given b

1. **Problem (a): Find the area enclosed by the curve $y = 4\cos 3x$, the x-axis, and the lines $x=0$ and $x=\frac{\pi}{6}$.** The area under a curve from $x=a$ to $x=b$ is given b

1. **Problem:** Find the total differential $dx$ if $x = y^3 z = \ln(z)$. 2. **Understanding the problem:** The total differential $dx$ of a function $x = f(y,z)$ is given by

1. The problem is to find the indefinite integral of the function $e^x$, which is written as $\int e^x \, dx$. 2. The formula for the integral of the exponential function $e^x$ is:

1. **Problem (a):** Determine $\int (1 - t)^2 \, dt$. 2. **Formula and rules:** Use the power rule for integration: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$.

1. **Problem Statement:** (a) Find the indefinite integral $\int (1 - t)^2 \, dt$.

1. Problem (a): Evaluate $\frac{dy}{dx}$ at $x=2.5$ for $y=\frac{2x^2+3}{\ln(2x)}$. Formula: Use the quotient rule for differentiation:

1. **State the problem:** Differentiate the function $y = (2x+3)^8$ with respect to $x$. 2. **Formula used:** We use the chain rule for differentiation, which states:

1. **State the problem:** Differentiate the function $$f(x) = (x+1)^6$$ with respect to $$x$$. 2. **Formula used:** Use the chain rule for differentiation. If $$f(x) = [g(x)]^n$$,

1. **Problem Statement for Q9:** Find the area between the piecewise function $$f(x) = \begin{cases} x^2 & 0 \leq x \leq 1 \\ 2 - x & 1 < x \leq 2 \end{cases}$$ and $$g(x) = x$$ ov

1. **State the problem:** We have the curve given by the function $$y = x^3 - x^2 - 2x$$ and we want to compute two things over the interval $$[-1, 2]$$: a) The definite integral u

1. Problem Q6: Find the area between the curves $T_1(n) = n^2 + 3n$ and $T_2(n) = 2n^2$ from $n=0$ to $n=5$. 2. Formula: The area between two curves $f(n)$ and $g(n)$ over $[a,b]$