∫ calculus

1. The problem is to solve an integral or equation using the method of change of variable (substitution). 2. Identify the integral or equation to solve. For example, consider the i

1. **State the problem:** Calculate the integral $$\int \frac{1}{\sqrt{x(x-1)}} \, dx$$. 2. **Rewrite the integrand:** Notice that $$\sqrt{x(x-1)} = \sqrt{x^2 - x}$$.

1. **Exercise 1a:** Calculate $I_a = \int \frac{dx}{x+3}$. Using the basic integral formula $\int \frac{dx}{x+c} = \ln|x+c| + C$, we get

1. The problem is to find the equation of the tangent plane to the surface given by $$z = e^x - y$$ at the point $$(2,2,1)$$. 2. First, find the partial derivatives of $$z$$ with r

1. **State the problem:** Find the equation of the tangent plane to the surface given by $$-7z - 3y - 8x = -19$$ at the point $$P(2,1)$$. 2. **Rewrite the surface equation to solve

1. The problem is to find the integral $$\int \frac{1}{t^2 - a^2} \, dt$$ where $a$ is a constant. 2. Recognize that the denominator can be factored using the difference of squares

1. **State the problem:** (a) Given the integral $$\int_a^{2a} (10 - 6x) \, dx = 1$$, find the two possible values of $$a$$.

1. **State the problem:** (a) Given that $$\int_a^{2a} (10 - 6x) \, dx = 1,$$ find the two possible values of $$a$$.

1. **Problem 7:** A street light is mounted on a 15-ft pole. A 6-ft man walks away at 5 ft/s. Find how fast the tip of his shadow moves when he is 40 ft from the pole. 2. Let $x$ b

1. Problem 7: A street light is mounted at the top of a 15-ft pole. A man 6 ft tall walks away from the pole at 5 ft/s. Find how fast the tip of his shadow is moving when he is 40

1. The problem is to find the rate of change of a function without using integration. 2. Rate of change typically refers to the derivative of a function, which measures how the fun

1. **State the problem:** We are given the derivative $f'(x)$ as a step function over the interval $[-2,5]$ and the initial value $f(-2)=3$. We need to recover the function $f(x)$

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,