∫ calculus

1. **State the problem:** We are given the integral equation $$\int_1^k \left( \frac{3}{\sqrt{x}} + 4 \right) dx = \frac{95}{4}$$ and need to find the positive constant $k$. 2. **R

1. **State the problem:** We have the curve $y = (x - 2)(x - 4)$ and a vertical line $x = k$ with $k > 4$. The total shaded area between the curve and the x-axis from $x=2$ to $x=4

1. Problem: Find the derivative of the function $f(x) = x^2 - 3x + 8$ using the definition of the derivative. State the domain of the function and its derivative. 2. Recall the def

1. The problem asks to estimate the values of the derivatives of the function $f$ at points $0$ through $7$ based on a given graph, and then sketch the graph of the 9th derivative

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

1. **State the problem:** Find the area bounded by the curve $y=4x^3$ between $x=1$ and $x=3$. 2. **Set up the integral:** The area under the curve from $x=1$ to $x=3$ is given by

1. **State the problem:** We need to find the area bounded by the curve $y=3x^2$ between $x=0$ and $x=6$. 2. **Set up the integral:** The area under the curve from $x=0$ to $x=6$ i

1. **Problem 29:** Find the derivative of \(y = (x^2 + 2)(x^4 + 4)^4\) using logarithmic differentiation. 2. Take the natural logarithm of both sides:

1. **Problem 1: Find the instantaneous velocity and acceleration of the drone at $t=5$ given $s(t) = 7t^3 - 3t^2 - 5t + 2$.** 2. The instantaneous velocity is the first derivative

1. **Problem (I):** Calculate $$\lim_{x \to 0} \frac{27 - (3 + x)^3}{x}$$ 2. **Step 1:** Expand the cube in the numerator:

1. Problem (a): Find $$\lim_{h \to 0} \frac{\sin\left(\frac{\pi}{2} + h\right) - 1}{h}$$. 2. Use the identity $$\sin\left(\frac{\pi}{2} + h\right) = \cos h$$.

1. **Problem 1:** Find the tangent line to $y=f(x)=\sqrt{x}+1$ at $a=3$. 2. First, compute $f(3)$:

1. The problem asks to determine the continuity of the function $g(x)$ at specific points based on the graph description. 2. At $x = -2$, the graph has a closed dot at $(-2,0)$ and

1. The problem describes the continuity and discontinuity of the function $g$ at specific points on the interval $[-2,3]$ based on the graph. 2. At $x = -2$, $g$ is continuous from

1. **State the problem:** We want to determine whether the series $$\sum_{n=2}^{\infty} \frac{n^2}{n^3 + 1}$$ converges or diverges using the integral test. 2. **Set up the integra

1. **State the problem:** Determine whether the series $$\sum_{n=1}^{\infty} \left(3^n + 1 \cdot 4^{-n}\right)$$ converges or diverges, and if it converges, find its sum. 2. **Rewr

1. **State the problem:** We need to evaluate the integral $$\int e^x \sqrt{81 - e^{2x}} \, dx.$$\n\n2. **Substitution:** Let $$u = e^x.$$ Then, $$du = e^x dx = u dx \implies dx =

1. **State the problem:** We want to show that the sum of the areas of the upper approximating rectangles under the curve $f(x) = 5x^2$ on the interval $[0,2]$ approaches $\frac{40

1. **Problem a:** Find an appropriate trigonometric substitution for $$\int (5x^2 - 3)^{3/2} \, dx$$ given $$x = \sqrt{\frac{3}{5}} \sec \theta$$. Step 1: Recognize the form inside

1. **State the problem:** We need to evaluate the integral $$\int 5 \sin^4(x) \cos^2(x) \, dx.$$\n\n2. **Rewrite the integral:** Express powers of sine and cosine in terms of power

1. **State the problem:** We need to find $\frac{dy}{dx}$ by implicit differentiation for the equation $$x^3 + y^3 = 5.$$\n\n2. **Differentiate both sides with respect to $x$:**\nU