∫ calculus

1. The Intermediate Value Theorem (IVT) states that if a function $f$ is continuous on a closed interval $[a,b]$, and $d$ is any number between $f(a)$ and $f(b)$, then there exists

1. **Statement:** Consider the function $f$ defined for $x>0$ by $$f(x)=2x \ln x -1.$$

1. **State the problem:** Evaluate the limit $$\lim_{x \to 6} \frac{x^2 - 32}{2x - 5}$$. 2. **Substitute the value:** First, substitute $x=6$ into numerator and denominator:

1. The problem asks to evaluate the limit $$\lim_{x \to 0} \frac{\sin(7x)}{x}$$. 2. We recognize a standard limit: $$\lim_{u \to 0} \frac{\sin u}{u} = 1$$.

1. **Problem 1:** Find the derivative with respect to $x$ of the function $D_x = x(4 - \frac{1}{2}x)$. 2. First, expand the expression inside the derivative:

1. State the problem: We have a piecewise function:

1. Solve the inequalities and equations: I. Solve \(|2x + 3| \leq |2x + 1|\)

1. Solve the inequalities and equations: I. Solve $$|2x + 3| \leq |2x + 1|$$

1. **State the problem:** We need to find the values of various limits and function values for the piecewise function $g(t)$ at specific points $t=0, 2,$ and $4$ from the graph pro

1. Problem: Evaluate $$\int_0^4 x^3 \sqrt{4x - x^2} \, dx$$ Step 1: Rewrite the integrand: $$\sqrt{4x - x^2} = \sqrt{-(x^2 -4x)} = \sqrt{4^2 - (x-2)^2}$$ but it's simpler to use su

1. **State the problem:** We want to evaluate the definite integral $$\int_0^4 x^3(4x - x^2)^{1/2} \, dx.$$\n\n2. **Simplify the integrand:** Notice that $$4x - x^2 = x(4 - x),$$ s

1. We need to integrate the function $$\int \csc^5 x \, dx$$. 2. Express $$\csc^5 x$$ as $$\csc^3 x \cdot \csc^2 x$$ to use identities.

1. The problem is to find the integral of the function $$dy = \frac{1}{\frac{1}{2^2} - x}$$ with respect to $x$. 2. Simplify the denominator: $$\frac{1}{2^2} = \frac{1}{4}$$, so th

1. The problem asks us to estimate $\sin\left(\frac{1}{2}\right)$ using linear approximation. 2. Linear approximation uses the formula $f(x) \approx f(a) + f'(a)(x - a)$ near a poi

1. We need to find the integral of $\cos^6 x\,dx$.\n2. Use the power-reduction formula for cosine powers: $\cos^6 x = \left(\cos^2 x\right)^3$.\n3. Recall that $\cos^2 x = \frac{1

1. **Problem f**: Find $\frac{d}{dx}(\sqrt{x} \sin x)$. Using product rule: $\frac{d}{dx} (u v) = u' v + u v'$ where $u = \sqrt{x} = x^{1/2}$ and $v = \sin x$.

1. We start with the given function: $$y = 7\cos(x^6) - \frac{1}{8}$$

1. We are given the function composition $F(x) = f(g(x))$. The problem asks for the derivative $F'(-3)$. 2. By the chain rule, the derivative of $F$ at $x$ is

1. **State the problem:** Find the equation of the tangent line to the curve $y = \sin(\sin(x))$ at the point $\left(4\pi, 0\right)$.\n\n2. **Find the derivative of $y$:$$\frac{dy}

1. Stating the problem: Evaluate the limit $$\lim_{x\to -1} \arctan\left(\frac{2x}{1-x^2}\right)$$. 2. Understand the expression inside the arctan function: $$\frac{2x}{1-x^2}$$.

1. **State the problem:** Given functions $f(x(t)) = t^2 - 3t + 2$ and $y(t) = t^3 - 4t^2 + 1$, find the derivatives $\dot{x}$ and $\dot{y}$ at $t=2$, and determine the nature (loc