∫ calculus

1. **Problem statement:** Given $\lim_{x \to c} f(x) = 10$ and $\lim_{x \to c} g(x) = -14$, evaluate the following limits using limit laws. 2. **Limit laws used:**

1. **Problem statement:** Given $$\lim_{x \to c} f(x) = 10$$ and $$\lim_{x \to c} g(x) = -14$$, evaluate $$\lim_{x \to c} (f(x) + 8g(x))$$ using limit laws. 2. **Limit laws used:**

1. The problem is to evaluate the indefinite integral $$\int (5x^3 - 2x + 4) \, dx$$. 2. The formula for integrating a power function is $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$

1. **Problem:** Find the derivative of $y = e^{\sin 4x}$. 2. **Formula and rules:** The derivative of $e^u$ with respect to $x$ is $e^u \cdot \frac{du}{dx}$ (chain rule).

1. The problem is to find the derivative of a function, but the function is not specified in the question. 2. The derivative of a function $f(x)$ is found using the definition or r

1. **Problem statement:** Given the function $m(q) = 2q e^{-q}$, we need to state the differentiation rules applicable and find the derivative at $q=0$. 2. **Differentiation rules:

1. We are asked to find the limit of the function $f(x)$ as $x$ approaches $-5$. 2. The limit of a function at a point $a$ is the value that $f(x)$ approaches as $x$ gets arbitrari

1. **State the problem:** Evaluate the integral $$\int (x+1)(3x^2+2) \, dx$$. 2. **Expand the integrand:** Use distributive property:

1. **State the problem:** We want to find the integral of the function $\sin^{-1}(\log x)$, which means we want to compute $$\int \sin^{-1}(\log x) \, dx.$$\n\n2. **Recall the form

1. **Problem Statement:** Calculate the integral of the logarithm function $\int \log(x) \, dx$. 2. **Formula and Rules:** We use integration by parts, where $\int u \, dv = uv - \

1. **State the problem:** Evaluate the definite integral $$\int_0^b \frac{3x^3 - x^2 + 2x - 4}{\sqrt{x^2 - 3x + 2}} \, dx$$ where the upper limit $b$ is unspecified.

1. The problem is to find the area of the region enclosed by the curves given by the function $y=...$ (please provide the full function expressions to proceed). 2. To find the area

1. **State the problem:** Find the area of the region enclosed by the curves. 2. **General formula:** The area between two curves $y=f(x)$ and $y=g(x)$ from $x=a$ to $x=b$ is given

1. **Problem statement:** We have the function $$f(t) = \frac{1}{20}t^3 - \frac{9}{10}t^2 + \frac{77}{20}t$$ which models the rate of change of water volume in a tank (in cubic met

1. The problem is to find the derivative of the function $f(x) = \tan^{-1}(2x)$.\n\n2. Recall the formula for the derivative of the inverse tangent function: $$\frac{d}{dx} \tan^{-

1. **State the problem:** Evaluate the definite integral $$\int_0^1 \frac{3x^3 - x^2 + 2x - 4}{\sqrt{x^2 - 3x + 2}} \, dx$$.

1. **State the problem:** We want to evaluate the integral $$\int \frac{\sqrt{1+\sqrt{e^{-x}}}}{\sqrt{e^x}} \, dx.$$\n\n2. **Rewrite the integrand:** Recall that $$\sqrt{e^x} = e^{

1. The problem is to evaluate the expression $$\int_3^9 x \, dx$$ which is the definite integral of the function $f(x) = x$ from $x=3$ to $x=9$. 2. The formula for the definite int

1. **Problem 1:** Determine which statement about the piecewise function \( f(x) = \begin{cases} 2 & x < 5 \\ 2x - 4 & x \geq 5 \end{cases} \) is true regarding differentiability a

1. The first problem asks which statement cannot be used to conclude that $f(0)$ exists. - (A) $\lim_{x \to 0} f(x)$ exists means the limit exists but does not guarantee $f(0)$ is

1. **Problem Statement:** We are given a function $f$ with a vertical tangent at $x=4$, a horizontal tangent at $x=5$, a jump discontinuity at $x=2$, and a removable discontinuity