∫ calculus

1. **State the problem:** Find the limit $$\lim_{x\to -4} \frac{x^2 - 15}{x^2 - 16}$$. 2. **Substitute the value directly:** Substitute $x = -4$ into the expression:

1. **State the problem:** We need to find the limit $$\lim_{x\to 3} \big(h(x)g(x)\big)$$ where functions $h$ and $g$ are given graphically. 2. **Recall the limit product rule:** If

1. **State the problem:** We have a particle moving along a horizontal axis with position function $s = f(t) = 5t^2 - 2$. We want to find the velocity $v$ at time $t = 10$ seconds.

1. **State the problem:** We are given the function $f(x) = \frac{1}{3}x^3 - x^2$ and need to find its derivative $f'(x)$. 2. **Recall the derivative rules:**

1. 문제: $\lim_{x \to 1} \frac{(x^2 + 1)(ax + b)}{x^4 - 1} = 2$를 만족시키는 실수 $a$, $b$를 구하시오. 2. 문제를 이해하기 위해 분모와 분자를 살펴봅니다.

1. **State the problem:** We need to find the limit $$\lim_{x \to 3} \frac{g(x)}{f(x)}$$ where functions $g$ and $f$ are given graphically. 2. **Analyze the behavior of $g(x)$ near

1. **State the problem:** Find the critical numbers of the function $$f(x) = 2x^3 - 15x^2 + 36x + 5$$. 2. **Recall the definition:** Critical numbers occur where the derivative $$f

1. **State the problem:** We need to find the limit $$\lim_{x \to -2} \frac{g(x)}{h(x)}$$ where functions $g$ and $h$ are given graphically. 2. **Analyze $g(x)$ near $x = -2$:** Fr

1. **State the problem:** We are given the function $f(x) = 2x^3$ and asked which statement about the behavior of $f$ at the point $(0,0)$ is true. 2. **Recall the definitions:**

1. **State the problem:** Find the derivative $\frac{d}{dx} \csc^{-1}(2x)$ at $x=\frac{3}{5}$.\n\n2. **Recall the formula:** The derivative of $y = \csc^{-1}(u)$ with respect to $x

1. **State the problem:** Find the derivative $y'$ if $y = \sec\left(e^{x^4}\right)$. 2. **Recall the formula:** The derivative of $\sec(u)$ with respect to $x$ is $\frac{d}{dx} \s

1. **State the problem:** Given the equation $xy=1$, and the rate of change $\frac{dx}{dt}=12$ when $x=2$ and $y=\frac{1}{2}$, find $\frac{dy}{dt}$.\n\n2. **Use implicit differenti

1. The problem states that the function $f(x)$ has horizontal asymptotes at $y = -4$ and $y = 6$, and a vertical asymptote at $x = 3$. 2. Horizontal asymptotes describe the behavio

1. The problem states that the function $f(x)$ has horizontal asymptotes at $y = -4$ and $y = 6$, and a vertical asymptote at $x = 3$. 2. Horizontal asymptotes describe the behavio

1. The problem asks for the limit of the function $f(x)$ as $x$ approaches 2, i.e., $\lim_{x \to 2} f(x)$.\n\n2. The limit $\lim_{x \to a} f(x)$ exists if and only if the left-hand

1. **State the problem:** Determine which statement is not true about the graph of $$f(x) = x^{\frac{4}{3}} + 4x^{\frac{1}{3}} + 2$$. 2. **Recall the function and its derivatives:*

1. **State the problem:** We want to find the limit $$\lim_{x \to 0^+} \ln\left(\frac{x}{\sin x}\right).$$\n\n2. **Recall important facts:** As $x \to 0$, $\sin x \approx x - \frac

1. The problem is to evaluate a limit where direct substitution results in an indeterminate form like $\frac{0}{0}$ or $\frac{\infty}{\infty}$.\n\n2. L'Hopital's theorem states tha

1. **State the problem:** We want to find the limit $$\lim_{x \to 0} \frac{1 - \cos^2 x}{5x}.$$\n\n2. **Recall the identity:** Note that $$1 - \cos^2 x = \sin^2 x$$ by the Pythagor

1. **State the problem:** Given the equation $$\sin^2 x + \cos^2 y = \frac{5}{4}$$ and the rate $$\frac{dy}{dt} = -\frac{\sqrt{3}}{2}$$ at $$x = \frac{2\pi}{3}$$ and $$y = \frac{3\

1. **State the problem:** We are given two functions $f$ and $g$, and their composition $h = g \circ f$. We know $f(2) = 4$, $f'(2) = \frac{1}{2}$, and $g'(4) = 10$. We want to fin