∫ calculus

1. **Stating the problem:** Evaluate the double integral

1. **State the problem:** Find the limit $$\lim_{x \to 9^-} \frac{1+x}{x-9}$$. 2. **Recall the limit concept:** We want to see what value the function approaches as $x$ gets closer

1. **State the problem:** Find the limit $$\lim_{x \to -\frac{3}{2}} \frac{1}{(2x + 3)^2}$$. 2. **Recall the formula and rules:** The limit of a function as $x$ approaches a value

1. **State the problem:** We need to find the gradient (slope) of the curve defined by the function $$y = x^2 - 2x + 3$$ at the point where $$x = 2$$. 2. **Recall the formula:** Th

1. The problem is to find the derivative of the function $\sqrt{u}$ with respect to $u$. 2. Recall the formula for the derivative of a power function: if $f(u) = u^n$, then $f'(u)

1. **State the problem:** Find the equation of the tangent line to the curve $y=3x^2+12x+12$ at the point where $x=1.1$. 2. **Recall the formula:** The slope of the tangent line to

1. **State the problem:** We are given the function $$f(x) = x^3 + 3x^2 + 5x + 18$$ and need to find all points $$(x,y)$$ on the graph where the tangent line has slope $$5.5$$. 2.

1. We are given the function $$y = \frac{x^2 + 4}{x - 3}$$ and asked to find its derivative $$\frac{dy}{dx}$$. 2. To differentiate a quotient, we use the Quotient Rule: $$\frac{d}{

1. **State the problem:** Find the derivative of the function $$y = \frac{8t - 7}{5t + 1}$$ using the quotient rule. 2. **Recall the quotient rule formula:** If $$y = \frac{u}{v}$$

1. **State the problem:** We want to find the limit $$\lim_{x \to +\infty} \frac{e^x + \ln x}{e^x - 1}$$

1. **State the problem:** Find the limit $$\lim_{x \to 3} \frac{x^2 - 8x + 15}{\sqrt{5x + 34} - 7}$$. 2. **Check direct substitution:** Substitute $x=3$:

1. **Problem:** Find $$\lim_{x \to \infty} \frac{8x^3 + 6x^2 - 10}{5x^3 + 7x + 8}$$. 2. **Formula and rule:** For limits at infinity of rational functions, divide numerator and den

1. **State the problem:** We have a piecewise function $$f(x) = \begin{cases} 26 - cx^2 & \text{if } x < 3 \\ d & \text{if } x = 3 \\ cx + 2 & \text{if } x > 3 \end{cases}$$

1. **State the problem:** Find the limit $$\lim_{x \to 0} 4x \sin\left(\frac{1}{3x}\right)$$. 2. **Recall the limit property:** The sine function is bounded between -1 and 1, i.e.,

1. **State the problem:** We have a function for the number of bicycles assembled per day after $d$ days of training:

1. **State the problem:** We are given two functions: $$g(x) = x^3 - 4x + 1$$

1. **State the problem:** We are given the difference quotient expression for a function $f$: $$f(x + h) - f(x) = -2hx^2 + 5hx + 7h^2x + 5h^2 + 3h^3$$

1. **State the problem:** We are given a piecewise function $f(x)$ and need to find for each $a = -5, -1, 2, 5$ the values of $f(a)$ and the limit $\lim_{x \to a} f(x)$ if they exi

1. The problem is to find the limit: $$\lim_{x \to 0} \frac{\ln(1 + 3x)}{3 \sin(x)}$$ 2. To solve this, we use the property that we can multiply and divide by the same expression t

1. **State the problem:** We are given the function $$f(x) = \frac{1}{2x^2} + x^2 - \sqrt{x} + 5$$ for $$x > 0$$ and need to find its derivative $$f'(x)$$. 2. **Recall derivative r

1. **State the problem:** We need to find the derivative of the function $$f(x) = \frac{x^3}{\sqrt{x}}$$ and express it in the form $$f'(x) = ax^n$$, then find the exact value of t