∫ calculus

1. The problem states: Given $y = \ln(2x^2 - 3y^2)$, find $\frac{dy}{dx}$. 2. Differentiate both sides with respect to $x$. Using implicit differentiation, the derivative of the le

1. The problem states that $g$ is a decreasing function with $g(4) = 6$ and $g'(4) = -2$. We need to find which statement about the derivative of the inverse function $g^{-1}$ is t

1. The problem states that $g$ is a decreasing function with $g(4) = 6$ and $g'(4) = -2$. We need to find which statement about the derivative of the inverse function $g^{-1}$ is t

1. The problem is to find the maximum and minimum points of a function. However, the function is not specified in the question. 2. To find maximum and minimum points, we typically

1. **Problem a:** Evaluate $$\lim_{x \to \frac{1}{2}} \frac{2x^2 + 5x - 3}{6x^2 - 7x + 2}$$ 2. **Step 1:** Substitute $x = \frac{1}{2}$ directly to check if the limit can be evalua

1. **Problem statement:** Given the function $g(x) = \int_{-3}^x f(t) \, dt$ where $f(x)$ is defined as a piecewise function with a linear segment from $(-3,-3)$ to $(-1,0)$ and a

1. **Problem:** Find the derivative of $f(x) = 5x^3 \sin x$. Step 1: Use the product rule: $\frac{d}{dx}[u v] = u' v + u v'$ where $u = 5x^3$ and $v = \sin x$.

1. **State the problem:** Given $g(x) = \int_{-3}^x f(t)\,dt$ where $f$ is a piecewise function defined on $[-3,3]$, find: (a) $g(1)$

1. The problem is to understand what a local extremum is in mathematics. 2. A local extremum refers to a point on a function's graph where the function reaches a local maximum or m

1. **State the problem:** We need to find the $x$-coordinates of the stationary points of the curve given by $$y = e^{-5x} \tan^2 x$$ for $$-\frac{\pi}{2} < x < \frac{\pi}{2}$$. 2.

1. **State the problem:** We need to find the $x$-coordinate of the stationary point of the curve given by $$y = \cos x \sin 2x$$ in the interval $$0 < x < \frac{1}{2} \pi$$, corre

1. **State the problem:** We need to find the $x$-coordinate of the stationary point of the curve given by $$y = \cos x \sin 2x$$ in the interval $$0 < x < \frac{1}{2} \pi$$, corre

1. **State the problem:** We have the curve defined by $$y = \frac{e^{\sin x}}{\cos^2 x}$$ for $$0 \leq x \leq 2\pi$$. We need to find $$\frac{dy}{dx}$$ and then find the $$x$$-coo

1. **State the problem:** Given parametric equations $x = te^{2t}$ and $y = t^2 + t + 3$, we need to (a) show that $\frac{dy}{dx} = e^{-2t}$ and (b) show that the normal to the cur

1. **State the problem:** Given the curve defined by the equation $$x^3 + 3x^2y - y^3 = 3,$$ we need to:

1. **State the problem:** We need to find the $x$-coordinate of the stationary point of the curve given by $$y = \cos x \sin 2x$$ in the interval $$0 < x < \frac{1}{2} \pi$$, corre

1. **State the problem:** Find the $x$-coordinates of the stationary points of the curve given by $$y = e^{-5x} \tan^2 x$$ for $$-\frac{\pi}{2} < x < \frac{\pi}{2}$$. 2. **Find the

1. **State the problem:** Differentiate the function $$f(x) = (2x^2 + 3)((x^5 - x + 2)^3)$$ with respect to $$x$$. 2. **Identify the rule:** This is a product of two functions, so

1. **State the problem:** Differentiate the function $$f(x) = \frac{x}{1-x} - \frac{3x^3}{4}$$ with respect to $$x$$. 2. **Differentiate the first term:** Use the quotient rule for

1. **State the problem:** Differentiate the function $$f(x) = \frac{\sqrt{3+x}}{\sqrt[3]{x^2-2}}$$ with respect to $$x$$. 2. **Rewrite the function using exponents:**

1. **Stating the problem:** We want to find the integral $$\int x^2 e^x \, dx$$. 2. **Method:** Use integration by parts, where we let: