∫ calculus

1. The problem is to find the derivative of the function $$f(x) = e^{x} x^{2}$$. 2. We recognize that $$f(x)$$ is a product of two functions: $$u(x) = e^{x}$$ and $$v(x) = x^{2}$$.

1. **State the problem:** We are given the equation $$y\sqrt{x+1} = 4$$ and asked to find $$\frac{dy}{dx}$$ using implicit differentiation. 2. **Rewrite the equation:** The equatio

1. **State the problem:** We have a circular wetted area with area $A$ expanding at a rate of $\frac{dA}{dt} = 4$ mm$^2$/s. We want to find how fast the radius $r$ is expanding, i.

1. **State the problem:** We are given that the volume $V$ of a sphere is increasing at a rate of $\frac{dV}{dt} = 7$ cm³/s. We need to find the rate of change of its surface area

1. The problem is to find the derivative of $y$ with respect to the independent variable $t$. 2. To proceed, we need the explicit function $y(t)$ or the relationship between $y$ an

1. **State the problem:** Find the absolute extreme values of the function $$f(x) = \ln(x + 2) + \frac{1}{x}$$ on the interval $$[1, 10]$$. 2. **Find the derivative:** To locate cr

1. **State the problem:** We need to find the derivative $\frac{dy}{dt}$ where $y = \left(e^{\cos(t+9)}\right)^4$. 2. **Rewrite the function:** We can write $y$ as $y = e^{4\cos(t+

1. **Evaluate the limits involving absolute value:** Given the function $$f(x) = \frac{1}{x} - \frac{1}{|x|}$$

1. The problem asks us to evaluate the infinite series \(\sum_{k=1}^{\infty} \left(\frac{1}{\sqrt{k+1}} - \frac{1}{\sqrt{k+2}}\right)\) or determine if it diverges. 2. Notice that

1. **State the problem:** We want to find a formula for the partial sum $$S_n = \sum_{k=1}^n \left( \frac{1}{\sqrt{k+1}} - \frac{1}{\sqrt{k+2}} \right)$$ and then calculate the lim

1. **State the problem:** We are given the sequence $$S_n = \frac{\sin((n+1)\pi)}{12n+11}$$ and asked to find the limit as $$n \to \infty$$. 2. **Analyze the numerator:** Note that

1. **State the problem:** We want to analyze the infinite series $$\sum_{k=0}^\infty \left( \sin\left( \frac{(k+1)\pi}{12k+11} \right) - \sin\left( \frac{k\pi}{12k-1} \right) \righ

1. **State the problem:** We want to find the value of the infinite sum $$\sum_{k=0}^\infty \left( \sin \left( \frac{(k+1)\pi}{12k+11} \right) - \sin \left( \frac{k\pi}{12k-1} \rig

1. The problem is to differentiate the product of two terms: $4 - x$ and $\sin\left(\frac{n \pi x}{4}\right)$.\n\n2. We treat $4 - x$ as one function, say $u = 4 - x$, and $\sin\le

1. We are asked to evaluate the definite integral $$\int_0^{0.6} \left(5.3125 x^3 - 1.6375 x^2 - 0.8 x + 0.692\right) dx.$$\n\n2. First, find the antiderivative of the integrand te

1. **State the problem:** Given the function $f(x) = \frac{x^2}{4 + x}$, find the second derivative $f''(x)$, then evaluate $f''(0)$ and $f''(9)$. 2. **Find the first derivative $f

1. **State the problem:** We need to find the local extrema (maximum and minimum) of the function $$f(x) = 2xe^{-2x}$$ using the first derivative test. 2. **Find the first derivati

1. **State the problem:** We need to find the intervals where the function $$f(x) = 2x^3 - 3x^2 - 72x + 16$$ is increasing or decreasing. 2. **Find the derivative:** The first deri

1. **Problem 2.1:** Given $k(x) = \frac{df}{dx}$, find the constant $C$ such that $$\int_{1}^{4} k(x) \, dx = f(4) + C.$$ Step 1: Recognize that $k(x)$ is the derivative of $f(x)$,

1. The problem asks to find the interval where the derivative $f'$ of the function $f$ is negative. 2. From the graph description, $f$ has a maximum near $x=1$ and crosses the x-ax

1. The problem asks us to analyze the curve of the function $f''(x)$ and determine which statements about the function $f$ are true. 2. Recall that $f''(x)$ is the second derivativ