∫ calculus

1. **State the problem:** Evaluate the integral $$\int_1^2 \frac{16x^4}{(2x-1)^2} \, dx$$. 2. **Rewrite the integral:** The integral is $$\int_1^2 \frac{16x^4}{(2x-1)^2} \, dx$$.

1. **State the problem:** Find the maximum area of a rectangle inscribed in the region bounded by the curve $y = 8 - x^3$, the positive $x$-axis, and the positive $y$-axis.

1. **State the problem:** Differentiate the function $y=\sqrt{2x+1}$ using first principles (definition of derivative). 2. **Recall the definition of derivative:**

1. The problem is to determine if there is more than one critical point for a given function. 2. A critical point occurs where the derivative of the function is zero or undefined.

1. **State the problem:** We need to determine which of the given limits does not yield an indeterminate form. 2. **Recall indeterminate forms:** Common indeterminate forms include

1. The problem asks which condition guarantees that the tangent line approximation at $x=5.25$ is an overestimate of $f(5.25)$. 2. Recall that the tangent line approximation at $x=

1. **State the problem:** We need to find the derivative $\frac{dy}{dt}$ of the function $$y = (t^2 + 6t + 6)(4t^2 + 3).$$ 2. **Formula used:** To differentiate a product of two fu

1. **State the problem:** Find the derivative of the function $$f(x) = 5x^8 + 5x^3 - 6x - 6$$. 2. **Recall the derivative rules:**

1. **State the problem:** Determine if the infinite series $$\sum_{n=1}^\infty \frac{5n}{5n + 4n^{-10000}}$$ converges. 2. **Simplify the general term:** The term inside the sum is

1. **State the problem:** Evaluate the definite integral $$\int_{0}^{3 \pi} 90^2 \sin \left( \frac{1}{6} \theta \right) d\theta.$$\n\n2. **Recall the formula:** The integral of $$\

1. **State the problem:** Evaluate the integral $$\int 6 \cos \sqrt{x} \, dx$$. 2. **Substitution:** Let $$t = \sqrt{x} = x^{1/2}$$, then $$x = t^2$$.

1. **State the problem:** We are given values for functions $f$ and $g$ and their derivatives at $x=7$: $f(7) = -5$, $f'(7) = 4$, $g(7) = 1$, $g'(7) = -2$.

1. The problem asks to find the derivative of the function $f(x)$ at the point where $x=1$, denoted as $f'(1)$. 2. The derivative at a point is the slope of the tangent line to the

1. **State the problem:** We want to derive the function $f(x) = a^x$ where $a$ is a positive constant and $a \neq 1$. 2. **Recall the definition of the derivative:** The derivativ

1. **State the problem:** We want to find the derivative of the function $f(x) = a^x$, where $a$ is a positive constant. 2. **Recall the formula for the derivative of an exponentia

1. **State the problem:** Find all relative extrema (relative maxima and minima) of the function $$f(x) = x^4 - 32x + 3$$. 2. **Formula and rules:** To find relative extrema, we fi

1. **Problem statement:** Find the area of the region $R$ bounded by $y = x - 1$ and $y = -(x-1)^2 + 2$.

1. **Problem 5:** Find the area of the region in the first quadrant enclosed by the graphs of $y=2-x^2$, $y=3\sin x$, and the y-axis. 2. The area between two curves $y=f(x)$ and $y

1. **Problem 1: Solve for $b$ given the integral equation** Given:

1. **Stating the problem:** We are given functions $f(x)$ and points $P(x_0, y_0)$, and we need to find the equations of the tangent lines to the graph of $f$ that pass through $P$

1. **State the problem:** Evaluate the definite integral $$\int_{e^{2}}^{e^{4}} \frac{1}{\sqrt{x}(1-\sqrt{x})} \, dx.$$\n\n2. **Substitution:** Let $$u = \sqrt{x} = x^{\frac{1}{2}}