∫ calculus

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}}

1. **State the problem:** Find the limit as $x \to \infty$ of the expression $$\frac{1}{\sqrt{4x^2 - 3x} - 2x}.$$\n\n2. **Rewrite the expression:** The expression is $$\frac{1}{\sq

1. The problem is to find the limit of a function as the variable approaches infinity. 2. The general idea is to analyze the behavior of the function when the input grows very larg

1. We are asked to find the integral of $\ln(1-\sqrt{x})\,dx$. 2. The integral is $\int \ln(1-\sqrt{x})\,dx$.

1. The problem asks to find the value(s) of $c$ in the interval $[0, \pi]$ such that the average value of the function $f(x) = 4 \sin(x) - 2 \sin(2x)$ equals $f(c)$. 2. We are give

1. We are asked to find the integral of $\sin^2(x) \cos(x) \, dx$. 2. The integral is $\int \sin^2(x) \cos(x) \, dx$.

1. Das Problem lautet: Überprüfen Sie die Ableitung der Funktion $f(x) = 2^{5x}$. 2. Die Ableitung einer Exponentialfunktion $a^{g(x)}$ ist gegeben durch die Kettenregel:

1. The problem is to find the Extrempunkte (extreme points) of a function, which are points where the function reaches a local maximum or minimum. 2. To find these points, we use t

1. **State the problem:** We are given the function $$g(x) = \int_0^x \sqrt{1 + 6t} \, dt$$ and asked to find its derivative $$g'(x)$$ using the Fundamental Theorem of Calculus. 2.

1. **Problem statement:** Evaluate the definite integrals of the piecewise linear function $f(x)$ given by the graph: - From $x=0$ to $x=2$, $f(x)=0$.

1. **Problem statement:** Evaluate the definite integrals of the piecewise linear function $f(x)$ given by: - $f(x) = 0$ for $0 \leq x \leq 2$

1. **Problem Statement:** Evaluate the definite integrals of the piecewise linear function $f(x)$ given the graph description: