∫ calculus

1. The problem is to differentiate the product of two terms: $4 - x$ and $\sin\left(\frac{n \pi x}{4}\right)$.\n\n2. We treat $4 - x$ as one function, say $u = 4 - x$, and $\sin\le

1. We are asked to evaluate the definite integral $$\int_0^{0.6} \left(5.3125 x^3 - 1.6375 x^2 - 0.8 x + 0.692\right) dx.$$\n\n2. First, find the antiderivative of the integrand te

1. **State the problem:** Given the function $f(x) = \frac{x^2}{4 + x}$, find the second derivative $f''(x)$, then evaluate $f''(0)$ and $f''(9)$. 2. **Find the first derivative $f

1. **State the problem:** We need to find the local extrema (maximum and minimum) of the function $$f(x) = 2xe^{-2x}$$ using the first derivative test. 2. **Find the first derivati

1. **State the problem:** We need to find the intervals where the function $$f(x) = 2x^3 - 3x^2 - 72x + 16$$ is increasing or decreasing. 2. **Find the derivative:** The first deri

1. **Problem 2.1:** Given $k(x) = \frac{df}{dx}$, find the constant $C$ such that $$\int_{1}^{4} k(x) \, dx = f(4) + C.$$ Step 1: Recognize that $k(x)$ is the derivative of $f(x)$,

1. The problem asks to find the interval where the derivative $f'$ of the function $f$ is negative. 2. From the graph description, $f$ has a maximum near $x=1$ and crosses the x-ax

1. The problem asks us to analyze the curve of the function $f''(x)$ and determine which statements about the function $f$ are true. 2. Recall that $f''(x)$ is the second derivativ

1. The problem states that the curve is given by $y = f(x)$ and the tangent line at any point $(x, y)$ on the curve is given by $y = g(x)$. 2. By definition, the tangent line $g(x)

1. The problem asks to identify which graph represents the derivative $f'(x)$ of the given function $y=f(x)$ shown in the top-right graph. 2. The original function $f(x)$ starts ne

1. The problem asks to determine the number of inflection points of the function $f(x)$ given the graph of its derivative $f'(x)$ on the interval $-1 \leq x \leq 3$. 2. Recall that

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. The problem states that the top-right graph represents the first derivative $f'(x)$ of a function $f(x)$ defined on $\mathbb{R}$. We need to identify which of the four given gra

1. The problem asks to determine whether the function defined on the interval $x \in [0,2[$ has an absolute minimum and/or maximum value. 2. From the graph description, the functio

1. The problem states that the given curve represents the first derivative $f'$ of a continuous function $f$ on $\mathbb{R}$. We need to identify the wrong statement among the opti

1. The problem states that $f'(3)$ is undefined and $f''(x) > 0$ for all $x \neq 3$ on the interval $[1,5]$. 2. Since $f''(x) > 0$ for $x \neq 3$, the function $f$ is concave upwar

1. The problem states that $f$ is continuous with $f(0) = 3$, $f'(2) = f'(-2) = 0$, and $f'(x) > 0$ for $-2 < x < 2$. 2. The condition $f'(2) = f'(-2) = 0$ means the slope of the t

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

1. **Problem statement:** We analyze the function $f$ defined on the interval $[1,5]$ based on the given graph and determine which statement among (a), (b), (c), and (d) is not cor

1. The problem asks to identify which graph (a, b, c, or d) could represent the original function $y = f(x)$ given that the opposite figure represents its first derivative $f'(x)$.

1. The problem asks which function among the options is increasing on the interval $]a,b[$ given that $f$ is a function defined on $[a,b]$ with values in $\overline{\mathbb{R}}$ an