∫ calculus

1. **Problem Statement:** Define the indefinite integral of a vector-valued function and provide an example. 2. **Definition:** The indefinite integral of a vector-valued function

1. **Problem Statement:** We are given the graph of the first derivative $y = f'(x)$ of a function $f(x)$, which is a parabola opening upwards crossing the x-axis at $x=0$. We need

1. The problem states that the given graph is the first derivative $f'(x)$ of a function $f(x)$, and we need to determine which graph could represent the general shape of $f(x)$. 2

1. **State the problem:** Find the derivative of the function $$f(t) = \sqrt[5]{t} - \frac{1}{\sqrt[5]{t}}$$. 2. **Rewrite the function using exponents:** Recall that $$\sqrt[5]{t}

1. The problem is to define the derivative of vector-valued functions and understand how to compute it with examples. 2. A vector-valued function is a function that outputs a vecto

1. Problem A: Find intervals of increase/decrease and local extrema for $y = x^3 - 2x^2 + x - 3$. 2. Use the first derivative test: $y' = 3x^2 - 4x + 1$.

1. **Problem:** Evaluate the derivative $\frac{d}{dx} \sec(2x + 1)$. 2. **Formula:** The derivative of $\sec u$ with respect to $x$ is $\frac{d}{dx} \sec u = \sec u \tan u \frac{du

1. **Problem:** Evaluate $\frac{d}{dx} \sec(2x+1)$. 2. **Formula:** The derivative of $\sec u$ with respect to $x$ is $\frac{d}{dx} \sec u = \sec u \tan u \frac{du}{dx}$.

1. The problem asks to calculate the derivative of the composition of functions $g(f(x))$ at $x=1$. 2. Given functions:

1. The problem asks to calculate the derivative of the composition of functions $g(f(x))$ and $f(x)$, denoted as $(gf)'(x)$, and then evaluate it at $x=1$. 2. Given functions:

1. **State the problem:** We need to find the partial derivative of the function $$Z = x^3 y^2 - \frac{y}{x^2} + \frac{1}{y}$$ with respect to $$x$$. 2. **Recall the rules:** When

1. The Mean Value Theorem (MVT) states that for a function $f$ continuous on the closed interval $[a,b]$ and differentiable on the open interval $(a,b)$, there exists at least one

1. **Problem Statement:** We need to identify which graph represents a continuous function $f$ such that $f(0) = 3$, $f'(2) = f'(-2) = 0$, and $f'(x) > 0$ for $-2 < x < 2$. 2. **Un

1. The problem asks which graph could represent the function $h(x) = f(x) - g(x)$ given the graphs of $f'(x)$ and $g'(x)$. 2. Recall that $f'(x)$ and $g'(x)$ are the derivatives of

1. Problem: Find $$\int \frac{1}{\sqrt{12 + 4x - x^2}} \, dx$$. 2. Rewrite the expression under the square root by completing the square:

1. The problem asks which function among the options is increasing on the interval $]a,b[$ given that $f:[a,b]\to \mathbb{R}^+$ and $f$ is represented by a curve. 2. We know $f(x)>

1. The problem asks us to find the derivative curve $f'(x)$ of the given function $y=f(x)$, which is a downward-opening parabola with vertex near $(0,3)$ and roots near $x=-2$ and

1. **Problem Statement:** We are given that for each $x \in [a,b]$, the first derivative $f'(x) < 0$ and the second derivative $f''(x) > 0$. We need to determine which curve among

1. **Stating the problem:** We are given a piecewise linear function defined on the interval $[0,2[$ with points $(0,2)$, $(1,2)$, and $(2,0)$, and the function continues horizonta

1. **Problem Statement:** Given two differentiable functions $f(x)$ and $g(x)$ on the interval $[a,b]$, determine which of the following functions is always increasing on $[a,b]$:

1. **Stating the problem:** We need to evaluate two definite integrals: