∫ calculus

1. **State the problem:** Find the limit $$\lim_{x \to -\infty} \left(9^{x-2} + 4\right).\n\n2. **Recall the properties of exponential functions:** For any base $a > 1$, $$\lim_{x

1. **State the problem:** We need to find the limit as $x$ approaches 2 of the function $x^2 - 3x + 5$. 2. **Recall the limit rule for polynomials:** For any polynomial function $f

1. **State the problem:** Find all values of $c$ in the interval $[-1,4]$ such that the Mean Value Theorem (MVT) holds for the function $f(x) = x^3 - 2x^2 - 3x - 6$. 2. **Recall th

1. **State the problem:** We want to estimate the value of $\sqrt{8.9}$ using local linear approximation. 2. **Formula and concept:** Local linear approximation uses the tangent li

1. **State the problem:** Differentiate implicitly the equation $$\tan(x^{2}y^{4}) = 3x + y^{2}$$ to find $$\frac{dy}{dx}$$. 2. **Recall the formula and rules:**

1. **State the problem:** Find the derivative of the function $f(x) = \sqrt{\frac{1}{x}}$ using the limit definition of the derivative. 2. **Recall the limit definition of the deri

1. **Problem Statement:** Determine whether the functions \(f(x) = \frac{\sin^2(2x)}{4x^5}\) (number 8) and \(f(x) = \frac{1 - \cos x}{1 + \sin x}\) (number 10) diverge to positive

1. The problem asks whether the numbers 8 and 10 diverge to positive or negative infinity. 2. Divergence to infinity typically applies to functions or sequences, not individual num

1. Problem: Find the behavior or limit of $f(x) = \frac{\sin 4x}{x}$ as $x \to 0$. Formula: Use the limit $\lim_{x \to 0} \frac{\sin ax}{x} = a$.

1. Problem 17: Given $F(x) = \int_0^{x^2} \sqrt{2t} + 2 \, dt$, find $F'(1)$. 2. Use the Fundamental Theorem of Calculus and chain rule: If $F(x) = \int_a^{g(x)} f(t) \, dt$, then

1. Problem: Evaluate $(f \circ g)'(1)$ given $f(1)=2$, $g(1)=2$, $f'(1)=3$, $g'(1)=2$, $f'(2)=1$, and $g'(2)=3$. Formula: The derivative of a composite function is given by the cha

1. **State the problem:** We need to find the limit $$\lim_{x \to 2} \frac{x^2 - 4}{x^2 + x - 6}.$$\

1. Problem: Find the limit $$\lim_{y \to 7} \frac{y^2 - 4y - 21}{3y^2 - 17y - 28}$$ 2. Formula and rules: To find limits of rational functions, first try direct substitution. If it

1. **Problem statement:** Find the limit $$\lim_{z \to 8} \frac{2z^2 - 17z + 8}{8 - z}$$. 2. **Formula and rules:** To find limits involving rational functions, first try direct su

1. Problem: Find the limit $$\lim_{y \to 7} \frac{y^2 - 4y - 21}{3y^2 - 17y - 28}$$ 2. Formula and rules: To find limits of rational functions, first try direct substitution. If it

1. **Problem:** Find the limit $$\lim_{x \to 2} (8 - 3x + 12x^2)$$ 2. **Formula and rules:** For polynomial functions, limits can be found by direct substitution.

1. The problem is to evaluate the definite integral $$\int_1^4 \frac{x^2}{x^3+2} \, dx$$. 2. To solve this integral, we use substitution. Let $$u = x^3 + 2$$.

1. The problem is to evaluate the definite integral $$\int_0^2 e^{-t} \, dt$$. 2. The formula for the integral of an exponential function is $$\int e^{at} \, dt = \frac{1}{a} e^{at

1. The problem is to find the derivative of the function $g(x) = \ln(7x + 2)$.\n\n2. The formula for the derivative of a natural logarithm function $\ln(f(x))$ is given by:\n$$\fra

1. **State the problem:** Find the derivative of the function $$y = (3x + 2)^4 (8x - 5)^2$$ using logarithmic differentiation. 2. **Recall the formula and rules:**

1. **Stating the problem:** We are given the function $$V(n) = c (1 - 2N n)^n$$ and asked to find its derivative $$V'(n)$$ and then compute $$\frac{V'(n)}{V(n)}$$. 2. **Formula and