If \(3 x^2 + 2 x y + y^2 = 1\), then \(\dfrac{d y}{d x} =\)

The table gives selected values for the derivative of a function \(g\) on \(-1 \leq x \leq 2\): \(g'(-1.0)=2\), \(g'(-0.5)=4\), \(g'(0)=3\), \(g'(0.5)=1\), \(g'(1.0)=0\), \(g'(1.5)=-3\), \(g'(2.0)=-6\). If \(g(-1) = -2\) and Euler's method with a step-size of \(1.5\) is used to approximate \(g(2)\), what is the resulting approximation?

What are all values of \(x\) for which the series \(\displaystyle\sum_{n=1}^{\infty} \dfrac{n 3^n}{x^n}\) converges?

If \(\dfrac{d}{d x} f(x) = g(x)\) and if \(h(x) = x^2\), then \(\dfrac{d}{d x} f(h(x)) =\)

If \(F'\) is a continuous function for all real \(x\), then \(\operatorname*{lim}\limits_{h \rightarrow 0} \dfrac{1}{h} \displaystyle\int_{a}^{a+h} F'(x) d x\) is

\(\displaystyle\int_{0}^{3} \dfrac{d x}{(1 - x)^2}\) is

Which of the following series converge to \(2\)? I. \(\displaystyle\sum_{n=1}^{\infty} \dfrac{2 n}{n + 3}\) II. \(\displaystyle\sum_{n=1}^{\infty} \dfrac{-8}{(-3)^n}\) III. \(\displaystyle\sum_{n=0}^{\infty} \dfrac{1}{2^n}\)

If the function \(f\) given by \(f(x) = x^3\) has an average value of \(9\) on the closed interval \([0, k]\), then \(k =\)

Which of the following integrals gives the length of the graph \(y = \sin(\sqrt{x})\) between \(x = a\) and \(x = b\), where \(0 < a < b\)?

Which of the following integrals represents the area enclosed by the smaller loop of the graph of \(r = 1 + 2 \sin \theta\)?

The third-degree Taylor polynomial about \(x = 0\) of \(\ln(1 - x)\) is

If \(\dfrac{d y}{d x} = y \sec^2 x\) and \(y = 5\) when \(x = 0\), then \(y =\)

If \(f\) is differentiable at \(x = a\), which of the following could be false?

A solid has a rectangular base that lies in the first quadrant and is bounded by the x- and y-axes and the lines \(x = 2\) and \(y = 1\). The height of the solid above the point \((x, y)\) is \(1 + 3 x\). Which of the following is a Riemann sum approximation for the volume of the solid?

[Calculator] A particle moves along the x-axis so that at any time \(t \geq 0\) its velocity is given by \(v(t) = \ln(t + 1) - 2 t + 1\). The total distance traveled by the particle from \(t = 0\) to \(t = 2\) is

[Calculator] If the function \(f\) is defined by \(f(x) = \sqrt{x^3 + 2}\) and \(g\) is an antiderivative of \(f\) such that \(g(3) = 5\), then \(g(1) =\)

[Calculator] Let \(g\) be the function given by \(g(x) = \displaystyle\int_{1}^{x} 100(t^2 - 3 t + 2) e^{-t^2} d t\). Which of the following statements about \(g\) must be true? I. \(g\) is increasing on \((1, 2)\). II. \(g\) is increasing on \((2, 3)\). III. \(g(3) > 0\)

For a series \(S\), let \(S = 1 - \dfrac{1}{9} + \dfrac{1}{2} - \dfrac{1}{25} + \dfrac{1}{4} - \dfrac{1}{49} + \dfrac{1}{8} - \dfrac{1}{81} + \dfrac{1}{16} - \dfrac{1}{121} + ... + a_n + ...\), where \(a_n = \begin{cases} \dfrac{1}{2^{(n-1) slash 2}} \text{if n is odd} \\ \dfrac{-1}{(n + 1)^2} \text{if n is even} \end{cases}\). Which of the following statements are true? I. \(S\) converges because the terms of \(S\) alternate and \(\operatorname*{lim}\limits_{n \rightarrow \infty} a_n = 0\). II. \(S\) diverges because it is not true that \(|a_{n+1}| < |a_n|\) for all \(n\). III. \(S\) converges although it is not true that \(|a_{n+1}| < |a_n|\) for all \(n\).

[Calculator] Let \(g\) be the function given by \(g(t) = 100 + 20 \sin\left(\dfrac{\pi t}{2}\right) + 10 \cos\left(\dfrac{\pi t}{6}\right)\). For \(0 \leq t \leq 8\), \(g\) is decreasing most rapidly when \(t =\)