∫ calculus

1. **State the problem:** We need to find the gradient of the tangent to the curve $y=2x^3+5$ at the point $P(1,7)$ using differentiation from first principles. 2. **Recall the for

1. **State the problem:** We have parametric equations $x(t) = 2 - 4 \sin t$ and $y(t) = 4 \cos t$ for $t$ in $[0, \frac{\pi}{2}]$. We want to find the volume generated when the cu

1. The problem is to find the integral of the function $y=\frac{1}{x^2+1}$.\n\n2. The formula for the integral of $\frac{1}{x^2+1}$ is a standard integral: $$\int \frac{1}{x^2+1} \

1. **Stating the problem:** We want to find the integral $$\int \frac{1}{1+\cos^4 t} \, dt$$ and analyze its behavior for graphing. 2. **Formula and approach:** There is no simple

1. Stating the problem: Evaluate the integral $$\int \frac{1}{1+\cos^\&} \, dx$$. 2. Clarification: The expression \(\cos^{\&}\) is unclear. Assuming it is a typo and you meant \(\

1. **State the problem:** Find the maximum value of the function. Since the function is not explicitly given, let's assume a general function $y=f(x)$ and explain the process to fi

1. **State the problem:** We need to find the derivative of the function $$f(x) = \frac{-x^{2} + 300x - 20000}{0.8}$$. 2. **Rewrite the function:** Since dividing by 0.8 is the sam

1. **State the problem:** We are given that when $x=2$, the rate of increase of $x$ with respect to time is $\frac{4}{5}$ times the rate of decrease of $y$ with respect to time. Th

1. **State the problem:** A spherical balloon's surface area is increasing at a rate of 8 cm²/s. We need to find how fast the volume is changing when the radius is 5 cm. 2. **Relev

1. **State the problem:** Evaluate the indefinite integral $$\int x \arctan(18x) \, dx$$. 2. **Recall the integration by parts formula:**

1. **State the problem:** Evaluate the double integral

1. The problem asks which slope field corresponds to the function $y = e^x$. 2. Recall that the slope field for a differential equation $\frac{dy}{dx} = f(x,y)$ shows small line se

1. **State the problem:** Use the Intermediate Value Theorem (IVT) to show there is a root of the equation $$e^x = 9 - 8x$$ in the interval $$(0,1)$$. 2. **Rewrite the equation:**

1. **State the problem:** We want to find the value of the constant $c$ such that the piecewise function $$f(x) = \begin{cases} cx^2 + 6x & \text{if } x < 3 \\ x^3 - cx & \text{if

1. **State the problem:** We need to find the points where the piecewise function $$f(x) = \begin{cases} x + 8 & x < 0 \\ e^x & 0 \leq x \leq 1 \\ 2 - x & x > 1 \end{cases}$$

1. We are asked to find the derivative of the function $$h(x) = \frac{x^2 + 6x + 9}{x + 3}$$. 2. First, recognize that the numerator can be factored: $$x^2 + 6x + 9 = (x + 3)^2$$.

1. The problem asks to identify the x-values where the function $f$ has a removable discontinuity. 2. A removable discontinuity occurs at a point where the function is not defined

1. **Énoncé du problème** : Trouver la formule générale pour $s_n$ de la fonction $f(x) = x^2 + 3x + 1$ sur l'intervalle $[0,1]$ en utilisant la somme de Riemann à gauche.

1. **State the problem:** Find the limit of the function $f(x)$ as $x \to \infty$ based on the given graph description. 2. **Analyze the graph behavior:** The graph has vertical as

1. **State the problem:** We need to evaluate the integral $$\int \frac{(x^2 + 2)(3x^2 + 2x)}{\sqrt[3]{x^3 + 2x + 1}} \, dx.$$\n\n2. **Rewrite the integral:** The numerator is the

1. **State the problem:** We are given the function $f(x) = 2x^2 - e^x$ defined on the interval $[0,1]$ and a point $x^* = 0.36$. We want to evaluate or analyze the function at thi