∫ calculus

1. **Problem:** Compute $$\lim_{x \to a} \frac{3f(x) + g(x)}{h(x)}$$ given $$\lim_{x \to a} f(x) = 5$$, $$\lim_{x \to a} g(x) = -3$$, and $$\lim_{x \to a} h(x) = -2$$. 2. **Formula

1. **Problem statement:** (a) Use the definition of the derivative to find the derivative of $f(x) = \sqrt{x} - 3$.

1. **State the problem:** Find the limit $$\lim_{x \to -4} \frac{|2x^2 + 16x + 32|}{x^3 + 64}$$. 2. **Factor the expressions:**

1. **Problem statement:** (a) Estimate the gradient of the curve $y = \frac{1}{x} + \frac{x}{2}$ at $x=2$ by drawing a tangent.

1. **State the problem:** Find the differential coefficient (derivative) of the function $f(x) = 4x^2 - 2x + 1$. 2. **Recall the formula:** The derivative of a function $f(x)$ with

1. **State the problem:** Find the first four nonzero terms of the Taylor series for the function $$f(x) = \frac{5}{1+x}$$ centered at $$a=2$$. 2. **Recall the Taylor series formul

1. **State the problem:** Find the first four nonzero terms of the Taylor series for $f(x) = 5xe^x$ centered at $a=0$. 2. **Recall the Taylor series formula:**

1. **Problem Statement:** Find the Taylor series for the function $f$ centered at $6$ given that

1. **Problem Statement:** Find the Maclaurin series for the function $f$ given that its $n$th derivative at zero is $f^{(n)}(0) = (n+1)!$ for $n=0,1,2,\ldots$.

1. **Problem statement:** Evaluate the indefinite integral $$\int \frac{\tan^{-1}(x)}{x} \, dx$$ as a power series and find the radius of convergence $R$. 2. **Recall the power ser

1. **Problem statement:** Evaluate the indefinite integral $$\int x^7 \ln(1+x) \, dx$$ as a power series and find the radius of convergence $R$. 2. **Recall the power series expans

1. **Problem statement:** Evaluate the indefinite integral $$\int \frac{t}{1 - t^5} \, dt$$ as a power series and find the radius of convergence $R$. 2. **Recall the geometric seri

1. **State the problem:** Find the power series representation for the function $$f(x) = x^8 \tan^{-1}(x^3)$$ and determine its radius of convergence $$R$$. 2. **Recall the power s

1. **State the problem:** Evaluate the definite integral $$\int_1^2 \left(\frac{1}{x^2} - 3x + 8\right) dx.$$\n\n2. **Recall the integral rules:**\n- The integral of $x^n$ is $$\fr

1. **State the problem:** Differentiate the function $$f(x) = x^2 e^{3x}$$ with respect to $$x$$. 2. **Recall the formula:** To differentiate a product of two functions, use the pr

1. Problem 17: Find $$\lim_{x \to a} \frac{x^2 - 6x^3 + 11x - 6}{x^3 + 4x^2 - 19x + 14}$$. We first factor numerator and denominator if possible to simplify the expression. 2. Fact

1. **State the problem:** Evaluate the limit $$\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$$. 2. **Recall the formula and rules:** Direct substitution gives $$\frac{2^2 - 4}{2 - 2} = \fra

1. **Problem Statement:** We are given a continuous function $y = x^2$ and a graph with points labeled A, B, C, D, E, G, H, J showing local maxima and minima. We need to explain lo

1. **Problem Statement:** Find the derivative of the exponential function $y = e^x$. 2. **Formula and Rules:** The derivative of the exponential function $e^x$ with respect to $x$

1. **State the problem:** Find the second derivative of the function $h(x) = 2^x$. 2. **Recall the formula for the derivative of an exponential function:** For $a^x$ where $a > 0$

1. **Problem Statement:** We are given the function $$f(x) = \frac{x^3 - x^2}{x - 1}$$ and want to understand the behavior of $$f(x)$$ as $$x$$ approaches 1, which introduces the c