∫ calculus

1. We are asked to find \(\lim_{x \to -1} f(x)\). Looking at the graph, the function on the interval including \(x=-1\) is a straight line segment from (-3, -1) to (-1, 3). 2. To f

1. **State the problem:** Find the limit of $f(x)$ as $x$ approaches 1, i.e., $\lim_{x \to 1} f(x)$, and then check continuity at $x = 1$ by comparing the left-hand limit, right-ha

1. **Problem:** Air is pumped into a spherical balloon at 5 cm³/min. Find the rate of change of the radius when diameter is 20 cm. Step 1: Volume of sphere is $$V=\frac{4}{3}\pi r^

1. **State the problem:** We need to find the limits of the function $f(x)$ approaching $x=-1$ from the left, from the right, and overall at $x=-1$. 2. **Find the left-hand limit $

1. We are asked to evaluate the limit \(\lim_{x \to 1} \left( \frac{1}{1-x} - \frac{3}{1-x^3} \right)\).\n\n2. First, notice that as \(x \to 1\), the denominators \(1-x\) and \(1-x

1. **Problem Statement:** Evaluate the limit $$\lim_{\theta \to 0} \frac{\sin^2 \theta}{1 - \cos \theta}$$. 2. **Use Trigonometric Identities:** Recall the Pythagorean identity and

1. **State the problem:** Find the derivative of the function $$y=\frac{2x^4 \tan x}{e^{2x}\sin x}$$ and simplify. 2. **Rewrite the function:** It helps to write the function as a

1. Let's start by stating the problem: find the derivative of $$y=\frac{2x^4 \tan x}{e^{2x} \sin x}$$. 2. We can use the quotient rule, which states that for $$y=\frac{u}{v}$$, $$y

1. **Stating the problem:** Find the limit \( \lim_{x \to 2} \frac{x^2 + 2x - 1}{x^2 - 4} \). 2. **Substitute \(x = 2\) directly:**

1. Evaluate \( \lim_{x \to 0} \frac{\sin bx}{x} \). Using the standard limit \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \), we write \( \frac{\sin bx}{x} = b \cdot \frac{\sin bx}{bx} \

1. **Stating the problem:** We want to evaluate the limit $$\lim_{x\to \frac{\pi}{2}} x \sec^2 x$$. 2. **Recall the definition:** $$\sec x = \frac{1}{\cos x}$$, so $$\sec^2 x = \fr

1. We are asked to evaluate the limit $$\lim_{x \to 3} \frac{x - 1}{x^2 - x - 6}$$. 2. First, let's factor the denominator:

1. **State the problem:** We need to find the limits of the function $$f(x) = \frac{1 - 2x}{x^2 - 1}$$ as $$x$$ approaches $$-1$$ from the left ($$x \to -1^-$$), from the right ($$

1. Evaluate the integral $$\int \frac{3y}{y^2 + 4} \; dy$$ Step 1: Recognize the form suitable for substitution. Let $$u = y^2 + 4$$.

1. Problem: Find $$\lim_{x \to 3} (2x + 1)$$ Solution: Substitute $x=3$ directly since the function is polynomial and continuous.

1. **Problem:** Given $u = f(2x - 3y, 3y - 4z, 4z - 2x)$, prove that $$\frac{1}{2} \frac{\partial u}{\partial x} + \frac{1}{3} \frac{\partial u}{\partial y} + \frac{1}{4} \frac{\pa

1. Use the definition of the derivative to find the derivatives. 1.a. Given $Q(t)=10+5t-t^2$, the definition of derivative is $$Q'(t)=\lim_{h\to0}\frac{Q(t+h)-Q(t)}{h}.$$ Calculate

1. The problem is to find the slope of the tangent line to the curve \(y=x^3+4x-7\) at the point where \(x=2\). 2. The slope of the tangent line at any point is given by the deriva

1. The user has input the expression $\frac{d}{dx}$ which signifies the derivative operator with respect to $x$. 2. This expression alone is not a complete problem but indicates di

1. **State the problem:** We want to find the derivative $\frac{dR}{dq}$ given the function $$R = q \sqrt{1000 - q^2}.$$\n\n2. **Rewrite the expression:** Express the square root a

1. Evaluate the integral $$\int \frac{3y}{5y^2 + 4} dy$$. Step 1: Substitute to simplify the integral. Let $$u = 5y^2 + 4$$, then $$du = 10y dy$$, so $$y dy = \frac{du}{10}$$.