∫ calculus

1. **State the problem:** Given the rate of growth of Rapunzel's hair as $$\frac{dL}{dt} = \frac{1}{5t}$$, find the length of hair grown between the 100th day and the 200th day. 2.

1. The problem gives the function $$y = 2x^4 - x^2$$ and asks to find the value of $$x$$ for which the derivative $$y' = 0$$. 2. Find the derivative of $$y$$ with respect to $$x$$.

1. **Constant Rule:** The derivative of a constant is zero. Example: If $f(x)=5$, then $f'(x)=0$.

1. **State the problem:** We need to find the first derivative of the function $$f(x) = x^3 - 5x + 2$$ and then evaluate it at $$x = 2$$. 2. **Find the first derivative:** Differen

1. The problem is to find the second derivative of the function given as $$f(x) = 3x^3 - 2x^2 + 4x - 8$$. 2. First, find the first derivative $f'(x)$ by differentiating each term:

1. The problem is to find the second derivative of the function given by the equation: $$x^3 - 5x^2 + x = 0$$

1. **State the problem:** Find the derivative of the function $$f(x) = \frac{x+5}{x^2-1}$$. 2. **Recall the quotient rule:** If $$f(x) = \frac{u(x)}{v(x)}$$, then $$f'(x) = \frac{u

1. We are asked to find the first derivative of the function $$y = x^2 (x+1)^3$$. 2. To differentiate, we recognize this as a product of two functions: $$u = x^2$$ and $$v = (x+1)^

1. Problem: Evaluate the integral $$\int_3^4 \frac{3x}{\sqrt{x-2}} \, dx$$ using the substitution $u = \sqrt{x-2}$. Step 1: Express $x$ in terms of $u$: $$x = u^2 + 2$$.

1. The problem is to explain the integration by parts formula: $$\int_a^b u(x)v'(x)\,dx = [u(x)v(x)]_a^b - \int_a^b u'(x)v(x)\,dx.$$\n\n2. Integration by parts is derived from the

1. **Problem statement:** Find the derivatives of each function using the chain rule. 2. **Part (a):** $y=\sin^2(x^2)$

1. We are asked to find the derivative of the function $$f(x) = \frac{1 - x}{2x}$$ at $$x = -1$$ using the definition of derivative. 2. The definition of derivative at point $$a$$

1. Evaluate the limit \(\lim_{x \to 1} \frac{x^{3} - 1}{x - 1}\). Start by recognizing that direct substitution yields \(\frac{1^3 - 1}{1 - 1} = \frac{0}{0}\), an indeterminate for

1. The problem is to evaluate the limit $$\lim_{x \to a} \frac{x^3 - 1}{x - 1}$$. 2. Direct substitution would give $$\frac{a^3 - 1}{a - 1}$$. However, if $a=1$, this becomes $$\fr

1. The problem is to find the derivative of a function. 2. Start by stating the function explicitly if known (e.g., $f(x)$). If the function is not provided, please specify it.

1. سنبدأ بحساب حدود التمرين 01 (limits). 1.1. لحساب $$\lim_{x \to +\infty} \frac{\sqrt{x^2 - 3}}{x + 1}$$

1. **Problem Statement:** Estimate the following limits using the graph of $y=f(x)$: a. $\lim_{x \to 0^-} f(x)$

1. The problem asks to interpret the double integral \(\int_a^b\int_c^d f(x)\,dx\). 2. Notice the inner integral \(\int_c^d f(x)\,dx\) is with respect to \(x\) from \(c\) to \(d\).

1. The problem is to graphically solve the equation $\sin x = \cos x$. 2. To find where $\sin x = \cos x$, we can rewrite this as $\sin x - \cos x = 0$.

1. Stating the problem: We want to minimize the function $$f(x_1,x_2) = (x_1 - \sqrt{5})^2 + (x_2 - \pi)^3 + 10$$

1. **Problem 1:** Determine if the function $$