∫ calculus

1. **State the problem:** We need to find $\frac{dz}{dt}$ where $z = \sin(x) \cos(y)$, $x = t$, and $y = \frac{1}{t}$. 2. **Formula and rules:** Use the Chain Rule for multivariabl

1. **State the problem:** We are given velocity graphs of two particles and asked to determine intervals when each particle is speeding up or slowing down. 2. **Recall the rule:**

1. State the problem.\nFind $\dfrac{dy}{dx}$ for the parametric equations $y=\dfrac{\theta-1}{\theta+1}$ and $x=\dfrac{\theta^2-1}{\theta^2+1}$.\n\n2. Use the parametric derivative

1. **State the problem:** Find the slope of the tangent line to the function $$f(x) = \frac{4x - 1}{x}$$ at $$x = 1$$. 2. **Recall the formula:** The slope of the tangent line at a

1. **State the problem:** We are given the limit $$\lim_{x \to 1} \frac{h(x) - h(1)}{\ln(x^2)} = 2$$ and asked to find the value of $$h'(1)$$ and analyze if $$h(1)$$ is an extremum

1. **Problem:** Compute $\lim_{x \to c} (f(x) + g(x))$ given $\lim_{x \to c} f(x) = 1$ and $\lim_{x \to c} g(x) = -1$. 2. **Formula:** The limit of a sum is the sum of the limits:

1. **State the problem:** We want to evaluate the limit $$\lim_{x \to 4} (x + 2)$$ using a table of values. 2. **Recall the limit concept:** The limit of a function as $$x$$ approa

1. **State the problem:** We need to find the integral $$\int \frac{1}{x^3 (x^2 + 1)^2} \, dx.$$\n\n2. **Formula and approach:** To integrate rational functions like this, we use p

1. **State the problem:** Find the limit \( \lim_{x \to 3} \frac{x^2 - 9}{x - 3} \). 2. **Recall the formula and rules:** The expression is a rational function. Direct substitution

1. **Problem:** Find $\frac{dy}{dx}$ using implicit differentiation for the equation: $$x + \sec(y) = \ln(y)$$

1. **State the problem:** We want to find the integral $$I = \int \ln(x^2 + 1) \, dx$$. 2. **Use integration by parts formula:** $$\int u \, dv = uv - \int v \, du$$.

1. The problem is to find the derivative $f'(x)$ of the function $$f(x) = \int_{e^x}^{2-x} \sin(t^2) \, dt$$

1. **Problem:** Evaluate the definite integral \( \int_1^2 \left( \frac{2}{x^3} + 3x \right) dx \). 2. **Formula and rules:** The definite integral of a sum is the sum of the integ

1. **State the problem:** We are given the function $f(x) = x^2 \sin x$ and want to understand its behavior. 2. **Formula and rules:** The function is a product of $x^2$ and $\sin

1. **Stating the problem:** We are given the function $$f(x) = \frac{\ln(x)}{e^x}$$ and asked to analyze it. 2. **Formula and rules:** The function is a quotient of two functions:

1. **State the problem:** Differentiate the function $$f(x) = (2x^3 + 5)(3x^2 + 5x)$$ with respect to $$x$$. 2. **Formula used:** Use the product rule for differentiation, which st

1. **State the problem:** We need to find the first derivative $\frac{dy}{dx}$ using implicit differentiation for the equation $$4x^2 - 2y^2 = 9.$$\n\n2. **Recall the formula and r

1. **State the problem:** We are given a function $f(p)$ representing the number of shoes sold at price $p$. We know $f(120) = 9000$ and $f'(120) = -60$. The revenue function is $R

1. Stated problem: Calculate the integral $$\int (1 - x) \sin x \, dx$$. 2. Formula and rules: Use integration by parts, where $$\int u \, dv = uv - \int v \, du$$.

1. **State the problem:** We are given functions $j(x)$ and $k(x)$ with their values and derivatives at $x=-1$, and a function $$f(x) = \frac{1 - k(x)}{j(x)^2 - 4}.$$ We need to fi

1. **State the problem:** We want to find the volume $V$ of the solid generated by rotating the region bounded by the curve $y = \frac{x^4}{2}$ and the line $y = 2$ between $x=0$ a