∫ calculus

1. The problem is to find the derivative of the function $F(t) = (\ln(t))^2 \cos(t)$.\n\n2. We apply the product rule for differentiation: if $F(t) = u(t) v(t)$, then $F'(t) = u'(t

1. The minimum point of the graph of a function $f(x)$ is the point where the function reaches its lowest value in a particular region. 2. To find the coordinates of the minimum po

1. The problem asks for the one-sided limit of the function $f(x)$ as $x$ approaches 2 from the right, i.e., $\lim_{x\to 2^+} f(x)$. 2. According to the graph description, there is

1. We are asked to find the one-sided limit $$\lim_{x\to 2^+} f(x)$$ from the graph of the function $$f$$. 2. The graph indicates a vertical asymptote at $$x=2$$, shown by the dash

1. The problem asks for the one-sided limit \(\lim_{x\to 2^+} f(x)\), which means the value that \(f(x)\) approaches as \(x\) approaches 2 from the right side (values greater than

1. The problem asks for the value of $$\lim_{x\to -2} f(x)$$ based on the given graph of the function $$f$$. 2. To find the limit as $$x$$ approaches $$-2$$, we examine the $$y$$-v

1. We are given multiple differential equations and need to analyze or solve each. 2. The first equation is $$\frac{dy}{dx} = \frac{1 + y}{2 + x}$$.

1. **Problem f:** Evaluate $$\lim_{x \to 0} \frac{\frac{1}{x-6} + \frac{1}{6}}{3x}$$. Step 1: Simplify the numerator.

1. The quotient rule is a formula used to find the derivative of a function that is the quotient of two differentiable functions.\n2. Suppose we have two functions $f(x)$ and $g(x)

1. The problem asks to differentiate the expression $\frac{y}{t}$ with respect to $t$. 2. We recognize this as a quotient, so we apply the quotient rule: if $f(t) = \frac{y}{t}$, t

1. The expression is \(\sqrt[n]{x}\int_a^b f(x)\,dx\). 2. Here, \(\sqrt[n]{x}\) represents the \(n\)-th root of \(x\), or \(x^{1/n}\).

1. Find the derivative of $y=(2x^{4}+5)(3x^{5}-8)$ directly using the product rule.\nProduct rule: $(fg)'=f'g+fg'$.\n$f=2x^{4}+5$, $g=3x^{5}-8$.\n$f'=8x^{3}$, $g'=15x^{4}$.\nSo, $y

1. The problem is to determine whether the statement \n\n"If $$\int_{-1}^{1} f(x)\,dx = 0$$ then $$f(x)$$ is odd"\n\nis true or false.\n\n2. Recall the property of odd functions: f

1. Problem: Differentiate the function $y=\frac{7}{x}$. Rewrite the function as $y=7x^{-1}$ to apply the power rule.

1. Calculați integralale definite: 1.a) Calculați $$\int_0^3 \frac{dx}{x^2 + 3x + 2}$$.

1. We need to check if the function $$f(x)=\frac{x^2+3x+12}{x-3}$$ is continuous at $$x=4$$. 2. First, check if $$f(4)$$ is defined:

1. Let's start by understanding what an integral is. 2. The integral of a function represents the area under the curve of that function between two points on the x-axis.

1. **Find the limit** $\lim_{x \to 0} \frac{2}{x}$. As $x$ approaches 0, the denominator approaches 0 which causes the fraction to grow without bound. From the right side, $\frac{2

1. **State the first problem:** Find $$\lim_{x \to 4} \frac{3x^2 - 5x}{x + 6}$$. **Step 1:** Substitute $x = 4$: $$\frac{3(4)^2 - 5(4)}{4 + 6} = \frac{3(16) - 20}{10} = \frac{48 -

1. **State the problem:** Evaluate the limit $$\lim_{(x,y)\to(2,-4)} \frac{x+4}{x^2y - xy + 4^2 - 4x}$$ by factorization. 2. **Substitute the point** $(2,-4)$ directly to check for

1. **State the problem:** Find the limit as $x \to 2$ and $y \to -4$ of the expression $$\frac{x+4}{x^2 y - x y + 4^2 - 4x}.$$\n\n2. **Rewrite the expression:** The denominator is