∫ calculus

1. **State the problem:** Find the third derivative $y^{(3)}$ of the function $$y = \frac{\lg(x - 1)}{x^3}$$ where $\lg$ denotes the base-10 logarithm. 2. **Recall the formula and

1. **Problem statement:** We have an isosceles triangle with base length $20\sqrt{3}$ cm. The two equal legs decrease at a rate of 3 cm/h. We want to find the rate at which the tri

1. Let's state the problem: Optimization involves finding the maximum or minimum value of a function, often subject to certain constraints. 2. The general approach uses the derivat

1. **State the problem:** Find all points on the curve defined by $$x^2 + y^2 + 4x - 2y = -1$$ where the tangent lines are (a) horizontal and (b) vertical. 2. **Rewrite the curve e

1. The problem is to find the derivative of a function, but the function is not specified. Please provide the function to differentiate.

1. **State the problem:** Find the derivative of the function $g(x) = 3e^x \cos x$. 2. **Formula used:** To differentiate a product of two functions, use the product rule:

1. **State the problem:** Find the equation of the tangent line to the curve $f(x) = 5e^x + 2\sin x$ at $x=0$. 2. **Recall the formula for the tangent line:** The equation of the t

1. **State the problem:** Find the derivative of the function $$y = \frac{5x^4 + 3x - 5 + 2}{x^3 + \pi x}$$. 2. **Recall the formula:** For a function $$y = \frac{u(x)}{v(x)}$$, th

1. **Problem Statement:** Find the equation of the tangent line to the curve at the given $x$ value. 2. **Formula and Rules:** The equation of the tangent line at $x=a$ is given by

1. **State the problem:** We are given the displacement function $s(t) = 4t^3 + 3t^2$ and need to find the instantaneous rate of change of displacement at $t=3$ seconds. This rate

1. **State the problem:** Find the $x$-value on the curve $y = x^2$ where the tangent line is parallel to the line $y = 2x - 4$. 2. **Identify the slope of the given line:** The li

1. The problem asks to find the value of $x$ where the instantaneous rate of change of the function $f(x) = x^4$ is equal to 32. 2. The instantaneous rate of change of a function a

1. **Problem statement:** Given the function $$y=\frac{x^2+4}{x^2-4}$$, find intervals of increase/decrease, concavity, and critical points. 2. **Find the first derivative:** Use t

1. **Problem Statement:** Find the intervals where the function $$y=\frac{x^2+4}{x^2-4}$$ is increasing or decreasing.

1. **Plantear el problema:** Queremos encontrar el área de la región sombreada delimitada por las curvas:

1. **State the problem:** Find the equation of the normal line to the curve $y = 3x^3 - 2x$ at the point $(1,1)$. 2. **Recall the formula:** The slope of the tangent line to the cu

1. **State the problem:** Find the equation of the normal line to the curve $y = 3x^3 - 2x$ at the point $(1,1)$. 2. **Recall formulas:**

1. Let's clarify the concepts of local minimum and absolute maximum. 2. A local minimum at a point means the function's value there is lower than all nearby points.

1. **Problem Statement:** Find the local maxima, local minima, absolute maximum, and absolute minimum of the function $f$ on the interval $[0,8]$ based on the given graph.

1. **Problem statement:** We have a piecewise function defined as: $$f(x) = \begin{cases} \frac{1}{x} + 4 & \text{if } x < 2 \\ 16 & \text{if } x = 2 \\ -2x^2 + 2x - 8 & \text{if }

1. **Stating the problem:** We are given a function $f$ with conditions involving $f^{-1}$, $f(x) + 3x > 0$, $f(3) = -8$, and another function $h(x) = (x^2 + 1) \cdot \ln(f(x) + 3x