∫ calculus

1. **State the problem:** We need to find the derivative $f'(x)$ of the function $$f(x) = \frac{x + 8}{x^2 - 9x + 2}.$$ 2. **Recall the formula:** For a function given by a quotien

1. **State the problem:** We need to find the derivative of the function $$y = (\sqrt{x} - 4)(x^2 + 17)$$ with respect to $$x$$. 2. **Recall the formula:** To differentiate a produ

1. **State the problem:** We have a particle moving with position function $$f(t) = t^3 - 15t^2 + 72t$$ for $$t \geq 0$$, where $$t$$ is time in seconds and $$s$$ is position in fe

1. **State the problem:** Find the derivative of the function $$f(x) = e^x \cosh(x)$$. 2. **Recall the formula:** To differentiate a product of two functions, use the product rule:

1. **State the problem:** Evaluate the limit $$\lim_{x \to \infty} \frac{2 + 9x^3}{(1 + x)(3 + x^2)}.$$ 2. **Rewrite the denominator:** Expand the denominator to understand its deg

1. **State the problem:** We want to find the derivative $G'(x)$ of the function $$G(x) = \int_{x^2}^{e^x} (\sin t)^4 \, dt.$$ 2. **Recall the Leibniz rule for differentiation unde

1. **State the problem:** We need to determine why the function $f(x) = \ln(x - 27)$ is discontinuous at $a = 27$. 2. **Recall the domain of the natural logarithm:** The function $

1. **State the problem:** We want to determine if the equation $$x^3 - 3x - 0.9 = 0$$ has at least one root in the interval $$(0,1)$$ using the Intermediate Value Theorem (IVT). 2.

1. **State the problem:** Find the derivative with respect to $x$ of the function $$f(x) = \frac{x^4}{2} + x^3.$$\n\n2. **Recall the derivative rules:**\n- The derivative of $x^n$

1. **State the problem:** Find the limit $$\lim_{x \to 1} \frac{x^2 + 4x - 5}{x^2 - 1}$$. 2. **Check direct substitution:** Substitute $x=1$ into numerator and denominator:

1. **State the problem:** We want to find the limit $$\lim_{x \to 3} \frac{\sqrt{x^2 + 7}}{x + 1}$$. 2. **Recall the limit rules:** If the function is continuous at the point, we c

1. **State the problem:** We are given a function $$P(x,y) = x^2 y^2 + 3x y^4$$ where $$x = f(r,t) = r^2 + e^r$$ and $$y = f(r,t) = 2t^2 r + \ln t$$.

1. **State the problem:** We want to evaluate the integral $$\int \frac{\sqrt{x^2 + 16}}{x^4} \, dx$$ using a trigonometric substitution. 2. **Choose the substitution:** Since the

1. **State the problem:** We need to evaluate the integral $$\int \frac{\sqrt{x^2 + 16}}{x^4} \, dx$$ using an appropriate trigonometric substitution. 2. **Identify the substitutio

1. **State the problem:** We are given the graph of the derivative $f'$ of a twice-differentiable function $f$ on the domain $(-9,9)$, with points of inflection at $x=-5$, $x=-1$,

1. **State the problem:** We are given a function $h$ with derivative $h'(x) = 4e^{\cos x}$ and asked to find the value of $x$ in $[-3,0]$ where the instantaneous rate of change of

1. **State the problem:** We have a function $g(x) = f(x^2 - x)$ where $f$ is twice differentiable. We know values of $f$, $f'$, and $g''(0)$, and we want to find $f''(0)$.

1. The problem involves analyzing a graph with key points: open circles at $$(-6, 5)$$ and $$(11, 0)$$, a minimum near $$(-1.5, -2)$$, and a maximum near $$(7, 7)$$. 2. Open circle

1. **State the problem:** Find the second derivative of the function $$f(x) = \tan^2 x$$. 2. **Recall the formula:** The first derivative of $$\tan x$$ is $$\sec^2 x$$. Using the c

1. **State the problem:** We are given the position function of a particle as $$s(t) = t^2 - 7t + 18$$ where $s(t)$ is in feet and $t$ is in seconds. We need to find the velocity f

1. **State the problem:** We want to find the region of convergence for the MacLaurin series of the function $$F(x) = \frac{x}{1 - x - x^2}$$. 2. **Recall the function's denominato