∫ calculus

1. **State the problem:** We need to determine the sign of the derivative $\frac{dy}{dx}$ at points A, B, C, D, and E on a curve. 2. **Recall the meaning of $\frac{dy}{dx}$:**

1. **State the problem:** Evaluate the integral $$\int (3x+5)^7 \, dx$$. 2. **Use substitution:** Let $$u = 3x + 5$$.

1. **State the problem:** We have a curve defined by the equation $$y = 7x - kx^2$$ where $k$ is a constant. We are told that when $x = 2$, the gradient (derivative) of the curve i

1. **State the problem:** Evaluate the integral $$\int (3x+5)^7 \, dx$$. 2. **Formula and rule:** Use the substitution method for integrals of the form $$\int (ax+b)^n \, dx$$.

1. **State the problem:** We have the curve given by the equation $$y = \frac{9}{2}x^2 - 3x + 1$$ and we want to find the coordinates of the point where the gradient (slope) is 33.

1. **State the problem:** We need to find the gradient of the tangent line $T$ to the curve $y = 5x^3 - 8x$ at the point $(-2, -24)$. 2. **Recall the formula:** The gradient of the

1. **State the problem:** Find the derivative $\frac{dy}{dx}$ of the function $$y = \frac{5x^7 - 3x^9}{2x}$$ and simplify the result. 2. **Rewrite the function:** Simplify the expr

1. **State the problem:** We have the function $$x = \frac{1}{6}t^3 - 5t$$ and need to find: a) A value of $$t$$ such that $$\frac{dx}{dt} = 27$$.

1. **State the problem:** Find the derivative $\frac{dy}{dx}$ of the function $$y = \frac{5x^7 - 3x^9}{2x}$$ and simplify the result. 2. **Rewrite the function:** Simplify the expr

1. **State the problem:** We are given the function $h = (4g + 7)^2$ and need to find the derivative $\frac{dh}{dg}$. 2. **Recall the formula:** To differentiate a function of the

1. **State the problem:** We have the function $x = \frac{1}{6}k^3 - 4k$.

1. **Problem statement:** Given the derivative of a curve $$\frac{dy}{dx} = a e^{1-x} - 3x^2$$ where $$a$$ is a constant, and the point $$(1,4)$$ on the curve has a gradient of 2.

1. **Problem:** Evaluate the limit \( \lim_{x \to 5} \frac{x + 3}{x - 1} \). 2. **Formula and rules:** For limits of rational functions where the denominator is not zero at the poi

1. Soal pertama meminta kita menggunakan hubungan rekursi fungsi gamma untuk menyederhanakan \(\Gamma\left(\frac{7}{2}\right)\) dan \(\Gamma\left(-\frac{7}{2}\right)\). 2. Hubungan

1. **Problem:** Given $x = t^2 - 1$ and $y = t^3 - 4$, find $\frac{d^2 y}{d x^2}$ at $t=1$. 2. **Step 1: Find $\frac{dy}{dt}$ and $\frac{dx}{dt}$.**

1. **Problem Statement:** Evaluate the integral $$\int_{-\pi}^{\pi} \sin mx \sin nx \, dx$$

1. **Problem Statement:** Calculate the integral $\int x^2 \, dx$. 2. **Formula Used:** The power rule for integration states that for any real number $n \neq -1$,

1. **Definition of a Function:** A function is a rule that assigns each input exactly one output. For example, $f(x) = x^2$ means for every $x$, you get one $y$ value. 2. **Types o

1. Stating the problem: Find the limit $$\lim_{x \to 0} \left(\sqrt{x^2 + 9} - 3\right) / x^2.$$\n\n2. Formula and rules: When direct substitution leads to an indeterminate form li

1. **State the problem:** We are given the function $$y(x) = \frac{12}{2\sqrt{x}} \left(2 - 8x + \sqrt{x} - 4\right)$$ and asked to find its derivative $$y'(x)$$. 2. **Simplify the

1. **State the problem:** Find the limit $$\lim_{x \to 0} \left(\sqrt{x^2 + 9} - 3\right) / x^2.$$\n\n2. **Recall the formula and approach:** When direct substitution leads to an i