∫ calculus

1. The problem involves finding the equation of the tangent line to the function $f(x) = \sqrt[3]{x}$ at $x=8$ and using it to approximate $\sqrt[3]{11}$. 2. Recall the formula for

1. **Problem 1:** Find the derivative $\frac{dy}{dx}$ of the implicit function given by $$x^2 y - 5y = 12$$ and evaluate it at the point $(1, -3)$. 2. **Step 1:** Differentiate bot

1. **State the problem:** A spherical balloon is inflated at a rate of $\frac{dV}{dt} = 200\pi$ ft³/min. We want to find how fast the radius $r$ is increasing, i.e., $\frac{dr}{dt}

1. **State the problem:** We have a right triangle with hypotenuse 13, base $x$, and angle $\theta$ opposite the base. Given $\frac{dx}{dt} = 2$ units/sec, find $\frac{d\theta}{dt}

1. **State the problem:** Evaluate the limit $$\lim_{h \to 0} \frac{\sin\left(\frac{\pi}{3} + h\right) - \sin\left(\frac{\pi}{3}\right)}{h}$$. 2. **Recall the formula:** This limit

1. The problem is to evaluate the limit $$\lim_{h \to 0} \frac{\sin\left(\frac{\pi}{3} + h\right) - \sin\left(\frac{\pi}{3}\right)}{h}$$ which represents the derivative of $\sin x$

1. The problem asks to evaluate the limit \( \lim_{x \to 3^+} [f(x)g(x)] \). 2. To solve limits involving products, use the property:

1. **State the problem:** Find the limit $$\lim_{x \to 3^+} [f(x) g(x)]$$ using the given graphs of $f(x)$ and $g(x)$. 2. **Analyze the graphs at $x=3^+$:**

1. **State the problem:** We need to find the derivative of the function $f(x) = \cos(4x)$ and then evaluate it at $x = \frac{\pi}{12}$. 2. **Recall the formula:** The derivative o

1. **Problem Statement:** Write an equation of the line tangent to the graph of $$y = x^3 + 3x^2 + 2$$ at its point of inflection. 2. **Find the point of inflection:** The point of

1. **State the problem:** Find the values of $x$ where the function $f(x) = x^4 - 18x^2$ has a relative minimum. 2. **Find the first derivative:**

1. **State the problem:** Find the derivative $y'$ of the function $$y = \sin^4(x^3) = (\sin(x^3))^4.$$\n\n2. **Recall the chain rule:** If $y = [u(x)]^n$, then $$y' = n[u(x)]^{n-1

1. **State the problem:** Find the derivative $f'(x)$ of the function $$f(x) = \frac{x^3}{\tan x}.$$\n\n2. **Recall the formula:** To differentiate a quotient $\frac{u}{v}$, use th

1. **State the problem:** Find the derivative $y'$ of the function $y = x^2 \cos x$. 2. **Recall the formula:** To differentiate a product of two functions, use the product rule:

1. **State the problem:** Evaluate the limit $$\lim_{x \to 0} \frac{\sqrt{x+1} - 2}{x - 3}$$. 2. **Recall the formula and rules:** When evaluating limits that result in indetermina

1. **State the problem:** Find $\frac{dy}{dx}$ when $y = (x^2 - 2x)^{\tan x}$ for $x > \sqrt{2}$. 2. **Use logarithmic differentiation:** Take natural logarithm on both sides:

1. The problem is to evaluate the definite integral $$\int_1^e \frac{1}{x} \, dx$$. 2. The formula for the integral of $$\frac{1}{x}$$ is $$\int \frac{1}{x} \, dx = \ln|x| + C$$, w

1. Evaluate \( \lim_{x \to 0} \frac{e^{2x} - 1}{x} \). The problem asks for the limit of the expression as \( x \) approaches 0.

1. Evaluate \(\lim_{x \to 0} \frac{e^{2x} - 1}{x}\). The problem asks for the limit of a function as \(x\) approaches 0.

1. **State the problem:** Evaluate the limit $$\lim_{x \to 0} \frac{\ln(1+x) - x}{x^2}$$ using L'Hôpital's Rule. 2. **Recall L'Hôpital's Rule:** If the limit results in an indeterm

1. **State the problem:** Find the limit $$\lim_{x \to 0} \frac{e^{2x} - 1}{x}$$. 2. **Recall the formula and rules:** The limit resembles the derivative definition of the exponent