∫ calculus

1. **State the problem:** We need to find the derivative of the function $$f(x) = 10\sqrt{x^4 + 19}$$ at the point $$x = 3$$. 2. **Recall the formula:** The derivative of $$f(x) =

1. **State the problem:** We need to find the third derivative $f'''(x)$ of the function $$f(x) = -x^2 - 2x^3 + 5x^4 + e^{-4x}$$ and then evaluate it at $x=0$. 2. **Recall the rule

1. **Problem (a):** Sketch the function \( f(x) = \begin{cases} 2, & x \geq 0 \\ -1, & x < 0 \end{cases} \) and determine if it is continuous at \( x=0 \). - The function is consta

1. **State the problem:** Find the limit as $x$ approaches $-\infty$ of the function $$\frac{2x^2 + 2x^2 + 1}{x^2 + 3}.$$\n\n2. **Simplify the expression:** Combine like terms in t

1. Differentiate $y = \ln(1 + \sin^2 x)$.\n\nStep 1: State the problem. We want to find $\frac{dy}{dx}$ for $y = \ln(1 + \sin^2 x)$.\n\nStep 2: Use the chain rule for differentiati

1. The problem is to understand why $e^{-\infty} = 0$. 2. Recall the exponential function $e^x$ where $e$ is approximately 2.71828.

1. The problem asks: Calculate the integral of a function as an example. 2. Let's find the integral of $f(x) = 2x$ over the interval $[0,3]$.

1. The problem asks: What is an integral used for? 2. An integral is a fundamental concept in calculus used to find the accumulation of quantities, such as areas under curves, tota

1. Yes, exactly! The $2 + 1$ refers to the exponent of $x$ in the expression. 2. When integrating a power of $x$, like $x^n$, the exponent $n$ is increased by 1 to become $n+1$.

1. Let's restate the question: why do we add 1 to the exponent in the integral rule $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$? 2. The $2 + 1$ in your example comes from the power

1. Let's start by stating the problem: understanding the integral rule in calculus. 2. The integral rule helps us find the antiderivative or the area under a curve of a function.

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