∫ calculus

1. **State the problem:** We need to find the derivative $\frac{dz}{dx}$ when $z = xe^{xi}$, where $i$ is the imaginary unit. 2. **Recall the formula:** To differentiate a product

1. **State the problem:** We need to find the derivative $\frac{dz}{dx}$ when $z = xe^{xj}$, where $j$ is a constant. 2. **Recall the formula:** To differentiate a product of two f

1. The problem involves understanding and differentiating logarithmic and trigonometric functions, as well as summations and integrals. 2. Key formulas and rules:

1. **State the problem:** We want to find the limit as $x$ approaches $+\infty$ of the expression $$\ln(2x - 1) - \ln(2x + 3) + \frac{1}{2}.$$\n\n2. **Recall the logarithm property

1. **Problem a:** Find the volume of the solid generated by rotating the region bounded by $y=\sqrt[3]{x}$ and $y=\frac{x}{4}$ in the first quadrant about the y-axis. 2. **Step 1:

1. **State the problem:** Show that

1) Problem: Find $$\lim_{x \to -\infty} \frac{3x}{x+2}$$ Step 1. For limits at infinity of rational functions, divide numerator and denominator by the highest power of $$x$$ in the

1. **Problem 1:** Find the limit $$\lim_{x \to 4} (-x^3 + 13x^2 - 56x + 83)$$ 2. **Problem 2:** Find the limit $$\lim_{x \to 3} \frac{2x^2 - x - 1}{x - 1}$$

1. **Problem Statement:** We are given a function $f$ with vertical asymptotes at $x = -5$ and $x = 5$. We want to determine which limit expressions agree with the behavior of the

1. Let's clarify the problem: You are asking whether a limit is unbounded (tends to infinity) or if the limit does not exist (none of the limits). 2. When evaluating limits, there

1. **State the problem:** We are given a function $f(x)$ with vertical asymptotes at $x = -4$ and $x = 5$. We want to determine which limit expressions agree with the graph's behav

1. **State the problem:** A spherical balloon's radius decreases at a constant rate of 15 cm/min. We need to find the rate at which the volume of air is removed when the radius is

1. **Problem Statement:** Determine which graphs satisfy the limit condition $$\lim_{x \to 3} g(x) = 5$$. This means as $x$ approaches 3, the values of $g(x)$ approach 5. 2. **Unde

1. Let's clarify the problem: you want to find the derivative of a function, which means finding the rate at which the function changes with respect to its variable. 2. The derivat

1. The problem states that the integral equation passes through the point $(-3, -\frac{11}{2})$. 2. To solve this, we need the integral equation or function to evaluate it at $x =

1. **State the problem:** Find the integral of the function given by its derivative $$\frac{9}{2}x^2 + 7x - 2$$. 2. **Recall the formula:** The integral (antiderivative) of a funct

1. **Problem:** Compute $$\lim_{x \to a} \frac{3f(x) + g(x)}{h(x)}$$ given $$\lim_{x \to a} f(x) = 5$$, $$\lim_{x \to a} g(x) = -3$$, and $$\lim_{x \to a} h(x) = -2$$. 2. **Formula

1. **Problem statement:** (a) Use the definition of the derivative to find the derivative of $f(x) = \sqrt{x} - 3$.

1. **State the problem:** Find the limit $$\lim_{x \to -4} \frac{|2x^2 + 16x + 32|}{x^3 + 64}$$. 2. **Factor the expressions:**

1. **Problem statement:** (a) Estimate the gradient of the curve $y = \frac{1}{x} + \frac{x}{2}$ at $x=2$ by drawing a tangent.

1. **State the problem:** Find the differential coefficient (derivative) of the function $f(x) = 4x^2 - 2x + 1$. 2. **Recall the formula:** The derivative of a function $f(x)$ with