∫ calculus

1. Find $\frac{dy}{dx}$ if $y = 4x^7 + 3 \cos 2x - \log x$. Step 1: Differentiate each term separately.

1. **State the problem:** We need to find the volume of the solid formed by rotating the region enclosed by the curve $y=e^{-x^2}$, the x-axis, and the vertical lines $x=-1$ and $x

1. **State the problem:** We need to find the volume of the solid formed by rotating the region enclosed by the curve $y = e^{-x^2}$, the x-axis, and the vertical lines $x = -1$ an

1. **Problem (a):** Find $$\int_2^4 (5x - 2)^{-\frac{3}{2}} \, dx$$ in exact form. 2. **Step 1:** Use substitution. Let $$u = 5x - 2$$, then $$du = 5 \, dx$$ or $$dx = \frac{du}{5}

1. **State the problem:** We need to find the first partial derivatives $f_x$, $f_y$, and $f_z$ of the function $$f(x,y,z) = e^{x^2 y} + \cos(xz) + y^2.$$\n\n2. **Find $f_x$: Parti

1. Soal pertama membahas nilai ekstrim dan titik belok fungsi $y=f(x)$ dengan titik $a$, $b$, $c$, dan $d$.\n- Ekstrim di $x=a$ dan $x=c$ berarti $f'(a)=0$ dan $f'(c)=0$.\n- Titik

1. The problem asks us to identify which graph among the options (a, b, c, d) could represent the second derivative $f''(x)$ of the function $f(x)$ shown in the top-right plot. 2.

1. **Problem statement:** We are given the graph of the first derivative $f'$ of a continuous function $f$ on $\mathbb{R}$ and asked to identify the wrong statement among the optio

1. **State the problem:** We are given that for the function $f$ on the interval $[a,b]$, the first derivative $f'(x) < 0$ and the second derivative $f''(x) > 0$ for all $x \in [a,

1. The problem asks to identify the graph of a continuous function $f$ such that: - $f(0) = 3$

1. **Problem:** Evaluate $$\int \frac{dx}{\sqrt{(x-\alpha)(\beta - x)}}$$ where $$\beta > \alpha$$. Step 1: Use substitution $$x = \alpha + (\beta - \alpha) \sin^2 \theta$$.

1. **State the problem:** We are given the function for the number of words remembered after $t$ days: $$w(t) = 100 \times (1 - 0.1t)^2, \quad 0 \leq t \leq 10.$$ We need to find t

1. The problem asks us to find and simplify the difference quotient $$\frac{f(a+h)-f(a)}{h}$$ for the function $$f(x) = \cos x$$. 2. Substitute the function into the difference quo

1. **State the problem:** We want to find the number of points where the function $$f(x) = \begin{cases} \min(1, x^2, x^3), & x < 1 \\ \min(x^3, 3x - 2), & x \geq 1 \end{cases}$$

1. Problem: Given the equation $x^2 + y^2 = 16$, find $\frac{\partial y}{\partial x}$. Step 1: Differentiate both sides with respect to $x$ implicitly.

1. **State the problem:** We have a piecewise function $$f(x) = \begin{cases} \frac{x^3 - a x^2 + 2}{x^2 - 3x + 2} & 0 < x < 1 \\ b^2 x^2 + b x + 1 & x = 1 \\ \left(1 + (\ln c) \ta

1. Problem: Given $f(x) = \frac{2x+3}{x-1}$, find its inverse, domain, range, and verify compositions. 1. Find $f^{-1}(x)$:

1. Problem: Find $\frac{dz}{dt}$ if $z = x^2 y + \sin y$, with $x = t^2$ and $y = \ln t$. Step 1: Express $z$ in terms of $t$ using given substitutions.

1. Find $f_x$ and $f_y$ if $f(x,y) = x^3 y^2 + 4x$. Step 1: Identify the function: $f(x,y) = x^3 y^2 + 4x$.

1. **State the problem:** Evaluate the triple integral $$\int_{\frac{\pi}{2}}^{5} \int_0^0 \int_0^0 r \sin \theta \sec^2 \phi \cos \theta \, dr \, d\theta \, d\phi.$$ 2. **Analyze

1. Find $f_x$ and $f_y$ if $f(x,y) = x^3 y^2 + 4x$. Step 1: Identify the function: $f(x,y) = x^3 y^2 + 4x$.