∫ calculus

1. **State the problem:** Find the $n^{th}$ order derivative of the function $$y = \sin(5x) \cdot \cos(3x).$$ 2. **Use product rule:** Since $y$ is a product of two functions, $u =

1. **State the problem:** We need to find the derivative $\frac{dy}{dx}$ of the function $$y = 4x^{8} + 4 \sqrt{x} - 5 + \frac{3}{x^{\frac{8}{5}}}.$$\n\n2. **Rewrite the function f

1. **State the problem:** Find the $n$th derivative of the function $$f(x) = \frac{1}{(x-1)(x-2)(x-3)}.$$\n\n2. **Rewrite the function:** We have $$f(x) = \frac{1}{(x-1)(x-2)(x-3)}

1. **State the problem:** We want to find the limit as $x$ approaches 0 of the expression $$\frac{a - \sqrt{a^2 - x^2}}{x}.$$ 2. **Understand the expression:** The numerator is $a

1. **Problem statement:** Find all local extrema, the global maximum, and the global minimum of the functions: c) $$f(x) = e^{-\frac{x^2}{2}}$$

1. **Problem statement:** Find all local extrema, the global maximum, and the global minimum of the function \(f(x) = 4 - |x - 3|\) on the domain \([-5, 5]\). 2. **Understand the f

1. **Problem statement:** Find all local extrema, global maximum, and global minimum of each function on the domain $[-5,5]$. ---

1. **Problem statement:** A woman wants to travel from point A to point C on opposite sides of a circular lake with radius $r=3$ km. She can walk along the shore at 8 km/h and row

1. Problem: Find $\lim_{x \to +\infty} \cos \left(\frac{1}{x}\right)$. As $x \to +\infty$, $\frac{1}{x} \to 0$. Since cosine is continuous,

1. Problem: Evaluate the limit $$\lim_{x \to 0} x$$. Since the function is simply $x$, as $x$ approaches 0, the value approaches 0.

1. **State the problem:** Find the area between the x-axis and the curve given by the function $$y = 4x - x^2$$. 2. **Identify the points of intersection:** The area between the cu

1. **State the problem:** Evaluate the definite integral $$\int_6^9 \frac{3\sqrt{x} - 2}{4\sqrt{x}} \, dx.$$\n\n2. **Rewrite the integrand:** Recall that $$\sqrt{x} = x^{\frac{1}{2

1. **State the problem:** We are given the derivative of a function $y = f(x)$ as $$\frac{dy}{dx} = 3x + A\sqrt{x}$$ where $A$ is a constant, and the curve has a stationary point a

1. We are asked to evaluate the integral $$\int (\cos 5x + 4 \sec^2 x + 8 e^{4x} + \frac{2}{x}) \, dx.$$\n\n2. We can split the integral into the sum of integrals:\n$$\int \cos 5x

1. **State the problem:** We are given the integral equation $$\int_1^k \left( \frac{3}{\sqrt{x}} + 4 \right) dx = \frac{95}{4}$$ and need to find the positive constant $k$. 2. **R

1. **State the problem:** We have the curve $y = (x - 2)(x - 4)$ and a vertical line $x = k$ with $k > 4$. The total shaded area between the curve and the x-axis from $x=2$ to $x=4

1. Problem: Find the derivative of the function $f(x) = x^2 - 3x + 8$ using the definition of the derivative. State the domain of the function and its derivative. 2. Recall the def

1. The problem asks to estimate the values of the derivatives of the function $f$ at points $0$ through $7$ based on a given graph, and then sketch the graph of the 9th derivative

1. **State the problem:** We have a function

1. **State the problem:** Find the area bounded by the curve $y=4x^3$ between $x=1$ and $x=3$. 2. **Set up the integral:** The area under the curve from $x=1$ to $x=3$ is given by

1. **State the problem:** We need to find the area bounded by the curve $y=3x^2$ between $x=0$ and $x=6$. 2. **Set up the integral:** The area under the curve from $x=0$ to $x=6$ i