Chapter 3

A function is given. Choose the alternative that is the derivative, \(\frac{d y}{d x}\), of the function. \(y=x^{5} \tan x\) (A) \(5 x^{4} \tan x\) (B) \(5 x^{4} \sec ^{2} x\) (C) \(5 x^{4}+\sec ^{2} x\) (D) \(5 x^{4} \tan x+x^{5} \sec ^{2} x\)

A function is given. Choose the alternative that is the derivative, \(\frac{d y}{d x}\), of the function. \(y=\frac{2-x}{3 x+1}\) (A) \(-\frac{7}{(3 x+1)^{2}}\) (B) \(\frac{5-6 x}{(3 x+1)^{2}}\) (C) \(-\frac{9}{(3 x+1)^{2}}\) (D) \(\frac{7}{(3 x+1)^{2}}\)

A function is given. Choose the alternative that is the derivative, \(\frac{d y}{d x}\), of the function. \(y=\sqrt{3-2 x}\) (A) \(\frac{1}{2 \sqrt{3-2 x}}\) (B) \(-\frac{1}{\sqrt{3-2 x}}\) (C) \(-\frac{4}{3}(3-2 x)^{3 / 2}\) (D) \(\frac{2}{3}(3-2 x)^{3 / 2}\)

A function is given. Choose the alternative that is the derivative, \(\frac{d y}{d x}\), of the function. \(y=\frac{2}{(5 x+1)^{3}}\) (A) \(-\frac{30}{(5 x+1)^{2}}\) (B) \(\frac{-30}{(5 x+1)^{4}}\) (C) \(\frac{-6}{(5 x+1)^{4}}\) (D) \(\frac{6}{(5 x+1)^{2}}\)

A function is given. Choose the alternative that is the derivative, \(\frac{d y}{d x}\), of the function. \(y=3 x^{2 / 3}-4 x^{1 / 2}-2\) (A) \(2 x^{1 / 3}-2 x^{-1 / 2}\) (B) \(3 x^{-1 / 3}-2 x^{-1 / 2}\) (C) \(\frac{9}{5} x^{5 / 3}-\frac{8}{3} x^{3 / 2}\) (D) \(2 x^{-1 / 3}-2 x^{-1 / 2}\)

A function is given. Choose the alternative that is the derivative, \(\frac{d y}{d x}\), of the function. \(y=2 \sqrt{x}-\frac{1}{2 \sqrt{x}}\) (A) \(x+\frac{1}{x \sqrt{x}}\) (B) \(\frac{4 x-1}{4 x \sqrt{x}}\) (C) \(\frac{1}{\sqrt{x}}+\frac{1}{4 x \sqrt{x}}\) (D) \(\frac{4}{\sqrt{x}}+\frac{1}{x \sqrt{x}}\)

A function is given. Choose the alternative that is the derivative, \(\frac{d y}{d x}\), of the function. \(y=\sqrt{x^{2}+2 x-1}\) (A) \(\frac{x+1}{y}\) (B) \(4 y(x+1)\) (C) \(\frac{1}{2 \sqrt{x^{2}+2 x-1}}\) (D) \(-\frac{x+1}{\left(x^{2}+2 x-1\right)^{3 / 2}}\)

A function is given. Choose the alternative that is the derivative, \(\frac{d y}{d x}\), of the function. \(y=\frac{x^{2}}{\cos x}\) (A) \(-\frac{2 x}{\sin x}\) (B) \(\frac{2 x \cos x-x^{2} \sin x}{\cos ^{2} x}\) (C) \(\frac{2 x \cos x+x^{2} \sin x}{\cos ^{2} x}\) (D) \(\frac{2 x \cos x-x^{2} \sin x}{\sin ^{2} x}\)

A function is given. Choose the alternative that is the derivative, \(\frac{d y}{d x}\), of the function. \(y=\ln \left(\frac{e^{x}}{e^{x}-1}\right)\) (A) \(x-\frac{e^{x}}{e^{x}-1}\) (B) \(\frac{1}{e^{x}-1}\) (C) \(-\frac{1}{e^{x}-1}\) (D) \(\frac{e^{x}-2}{e^{x}-1}\)

A function is given. Choose the alternative that is the derivative, \(\frac{d y}{d x}\), of the function. \(y=\tan ^{-1} \frac{x}{2}\) (A) \(\frac{4}{4+x^{2}}\) (B) \(\frac{1}{2 \sqrt{4-x^{2}}}\) (C) \(\frac{2}{\sqrt{4-x^{2}}}\) (D) \(\frac{2}{x^{2}+4}\)

A function is given. Choose the alternative that is the derivative, \(\frac{d y}{d x}\), of the function. \(y=\ln (\sec x+\tan x)\) (A) \(\sec x\) (B) \(\frac{1}{\sec x}\) (C) \(\frac{1}{\sec x+\tan x}\) (D) \(-\frac{1}{\sec x+\tan x}\)

A function is given. Choose the alternative that is the derivative, \(\frac{d y}{d x}\), of the function. \(y=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}\) (A) 0 (B) 1 (C) \(\frac{4}{\left(e^{x}+e^{-x}\right)^{2}}\) (D) \(\frac{1}{e^{2 x}+e^{-2 x}}\)

A function is given. Choose the alternative that is the derivative, \(\frac{d y}{d x}\), of the function. \(y=\ln \left(\sqrt{x^{2}+1}\right)\) (A) \(\frac{1}{\sqrt{x^{2}+1}}\) (B) \(\frac{2 x}{\sqrt{x^{2}+1}}\) (C) \(\frac{1}{2\left(x^{2}+1\right)}\) (D) \(\frac{x}{x^{2}+1}\)

A function is given. Choose the alternative that is the derivative, \(\frac{d y}{d x}\), of the function. \(y=\sin \left(\frac{1}{x}\right)\) (A) \(\cos \left(\frac{1}{x}\right)\) (B) \(\cos \left(-\frac{1}{x^{2}}\right)\) (C) \(-\frac{1}{x^{2}} \cos \left(\frac{1}{x}\right)\) (D) \(-\frac{1}{x^{2}} \sin \left(\frac{1}{x}\right)+\frac{1}{x} \cos \left(\frac{1}{x}\right)\)

A function is given. Choose the alternative that is the derivative, \(\frac{d y}{d x}\), of the function. \(y=\frac{1}{2 \sin 2 x}\) (A) \(-\csc 2 x \cot 2 x\) (B) \(\frac{1}{4 \cos 2 x}\) (C) \(-4 \csc 2 x \cot 2 x\) (D) \(-\csc ^{2} 2 x\)

A function is given. Choose the alternative that is the derivative, \(\frac{d y}{d x}\), of the function. \(y=e^{-x} \cos 2 x\) (A) \(-e^{-x}(\cos 2 x+2 \sin 2 x)\) (B) \(e^{-x}(\sin 2 x-\cos 2 x)\) (C) \(2 e^{-x} \sin 2 x\) (D) \(-e^{-x}(\cos 2 x+\sin 2 x)\)

A function is given. Choose the alternative that is the derivative, \(\frac{d y}{d x}\), of the function. \(y=\sec ^{2}(x)\) (A) \(2 \sec x\) (B) \(2 \sec x \tan x\) (C) \(2 \sec ^{2} x \tan x\) (D) \(\tan x\)

A function is given. Choose the alternative that is the derivative, \(\frac{d y}{d x}\), of the function. \(y=x(\ln x)^{3}\) (A) \(\frac{3(\ln x)^{2}}{x}\) (B) \(3(\ln x)^{2}\) (C) \(-3 x(\ln x)^{2}+(\ln x)^{3}\) (D) \((\ln x)^{3}+3(\ln x)^{2}\)

A function is given. Choose the alternative that is the derivative, \(\frac{d y}{d x}\), of the function. \(y=\frac{1+x^{2}}{1-x^{2}}\) (A) \(-\frac{4 x}{\left(1-x^{2}\right)^{2}}\) (B) \(\frac{4 x}{\left(1-x^{2}\right)^{2}}\) (C) \(-\frac{4 x^{3}}{\left(1-x^{2}\right)^{2}}\) (D) \(\frac{2 x}{1-x^{2}}\)

A function is given. Choose the alternative that is the derivative, \(\frac{d y}{d x}\), of the function. \(y=\sin ^{-1} x-\sqrt{1-x^{2}}\) (A) \(\frac{1}{2 \sqrt{1-x^{2}}}\) (B) \(\frac{2}{\sqrt{1-x^{2}}}\) (C) \(\frac{1+x}{\sqrt{1-x^{2}}}\) (D) \(\frac{x^{2}}{\sqrt{1-x^{2}}}\)

\(Y\) is a differentiable function of \(x\) Choose the alternative that is the derivative \(\frac{d y}{d x}\). \(x^{3}-y^{3}=1\) (A) \(x\) (B) \(3 x^{2}\) (C) \(\frac{x^{2}}{y^{2}}\) (D) \(\frac{3 x^{2}-1}{y^{2}}\)

\(Y\) is a differentiable function of \(x\) Choose the alternative that is the derivative \(\frac{d y}{d x}\). \(x+\cos (x+y)=0\) (A) \(\csc (x+y)-1\) (B) \(\csc (x+y)\) (C) \(\frac{x}{\sin (x+y)}\) (D) \(\frac{1-\sin x}{\sin y}\)

\(Y\) is a differentiable function of \(x\) Choose the alternative that is the derivative \(\frac{d y}{d x}\). \(\sin x-\cos y-2=0\) (A) \(-\cot x\) (B) \(\frac{\cos x}{\sin y}\) (C) \(-\csc y \cos x\) (D) \(\frac{2-\cos x}{\sin y}\)

If \(x=t^{2}+1\) and \(y=2 t^{3},\) then \(\frac{d y}{d x}=\) (A) \(3 t\) (B) \(6 t^{2}\) (C) \(\frac{6 t^{2}}{\left(t^{2}+1\right)^{2}}\) (D) \(\frac{2 t^{4}+6 t^{2}}{\left(t^{2}+1\right)^{2}}\)

If \(f(x)=x^{4}-4 x^{3}+4 x^{2}-1,\) then the set of values of \(x\) for which the derivative equals zero is (A) \\{1,2\\} (B) \\{0,-1,-2\\} (C) \\{-1,2\\} (D) \\{0,1,2\\}

If \(f(x)=\ln \left(x^{3}\right),\) then \(f^{\prime \prime}(3)\) is (A) \(-\frac{1}{3}\) (B) -1 (C) -3 (D) 1

If a point moves on the curve \(x^{2}+y^{2}=25,\) then, at \((0,5), \frac{d^{2} y}{d x^{2}}\) is (A) 0 (B) \(\frac{1}{5}\) (C) -5 (D) \(-\frac{1}{5}\)

If \(x=t^{2}-1\) and \(y=t^{4}-2 t^{3},\) then, when \(t=1, \frac{d^{2} y}{d x^{2}}\) is (A) 1 (B) -1 (C) 3 (D) \(\frac{1}{2}\)

If \(f(x)=5^{x}\) and \(5^{1.002} \simeq 5.016,\) which is closest to \(f^{\prime}(1) ?\) (A) 0.016 (B) 5.0 (C) 8.0 (D) 32.0

If \(y=e^{x}(x-1),\) then \(y^{\prime \prime}(0)\) equals (A) -2 (B) -1 (C) 1 (D) \(e\)

If \(x=e^{\theta} \cos \theta\) and \(y=e^{\theta} \sin \theta,\) then, when \(\theta=\frac{\pi}{2}, \frac{d y}{d x}\) is (A) 1 (B) 0 (C) \(e^{\pi / 2}\) (D) -1

If \(x=\cos t\) and \(y=\cos 2 t,\) then \(\frac{d^{2} y}{d x^{2}}(\sin t \neq 0)\) is (A) \(4 \cos t\) (B) 4 (C) -4 (D) \(-4 \cot t\)

\(\lim _{h \rightarrow 0} \frac{(1+h)^{6}-1}{h}\) is (A) 0 (B) 1 (C) 6 (D) nonexistent

\(\lim _{h \rightarrow 0} \frac{\sqrt[3]{8+h}-2}{h}\) is (A) 0 (B) \(\frac{1}{12}\) (C) 1 (D) 192

\(\lim _{h \rightarrow 0} \frac{\ln (e+h)-1}{h}\) is (A) 0 (B) \(\frac{1}{e}\) (C) 1 (D) \(e\)

\(\lim _{x \rightarrow 0} \frac{\cos x-1}{x}\) is (A) -1 (B) 0 (C) 1 (D) \(\infty\)

If \(f(x)=\left\\{\begin{array}{ll}\frac{4 x^{2}-4}{x-1}, & x \neq 1 \\ 4, & x=1\end{array},\right.\) which of the following statements is(are) true? I. \(\lim _{x \rightarrow 1} f(x)\) exists II. \(f\) is continuous at \(x=1\) III. \(f\) is differentiable at \(x=1\) (A) none (B) I only (C) I and II only (D) I, II, and III

If \(g(x)=\left\\{\begin{array}{ll}x^{2}, & x \leq 3 \\ 6 x-9, & x>3\end{array},\right.\) which of the following statements is (are) true? I. \(\lim _{x \rightarrow 3} g(x)\) exists II. \(g\) is continuous at \(x=3\) III. \(g\) is differentiable at \(x=3\) (A) I only (B) II only (C) I and II only (D) I, II, and III

The function \(f(x)=x^{2 / 3}\) on [-8,8] does not satisfy the conditions of the Mean Value Theorem because (A) \(f(0)\) is not defined (B) \(f(x)\) is not continuous on [-8,8] (C) \(f(x)\) is not defined for \(x<0\) (D) \(f^{\prime}(0)\) does not exist

If \(f(x)=2 x^{3}-6 x,\) at what point on the interval \(0 \leqslant x \leqslant \sqrt{3},\) if any, is the tangent to the curve parallel to the secant line on that interval? (A) 1 (B) \(\sqrt{2}\) (C) 0 (D) nowhere

If \(h\) is the inverse function of \(f\) and if \(f(x)=\frac{1}{x},\) then \(h^{\prime}(3)=\) (A) -9 (B) \(-\frac{1}{9}\) (C) \(\frac{1}{9}\) (D) 9

\(\lim _{x \rightarrow \infty} \frac{e^{x}}{x^{50}}\) equals (A) 0 (B) 1 (C) \(\frac{1}{50 !}\) (D) \(\infty\)

\(\lim _{x \rightarrow 0} \frac{\sin 2 x}{x}\) is (A) 1 (B) 2 (C) \(\frac{1}{2}\) (D) 0

\(\lim _{x \rightarrow 0} \frac{\sin 3 x}{\sin 4 x}\) is (A) 1 (B) \(\frac{4}{3}\) (C) \(\frac{3}{4}\) (D) 0

\(\lim _{x \rightarrow 0} \frac{1-\cos x}{x}\) is (A) 0 (B) 1 (C) 2 (D) \(\infty\)

\(\lim _{x \rightarrow 0} \frac{\tan \pi x}{x}\) is (A) 0 (B) 1 (C) \(\pi\) (D) \(\infty\)

\(\lim _{x \rightarrow \infty} x^{2} \sin \frac{1}{x}\) (A) is 1 (B) is 0 (C) is \(\infty\) (D) oscillates between -1 and 1

\(\lim _{x \rightarrow 0} \frac{\sec x-\cos x}{x^{2}}\) \((\mathrm{A})=0\) \((B)=\frac{1}{2}\) \((C)=1\) \((\mathrm{D})=2\)

A pair of equations that represent a curve parametrically is given. Choose the alternative that is the derivative \(\frac{d y}{d x}\). \(x=\cos ^{3} \theta \quad\) and \(\quad y=\sin ^{3} \theta\) (A) \(-\cot \theta\) (B) \(\cot \theta\) (C) \(-\tan \theta\) (D) \(-\tan ^{2} \theta\)

A pair of equations that represent a curve parametrically is given. Choose the alternative that is the derivative \(\frac{d y}{d x}\). \(x=1-e^{-t} \quad\) and \(\quad y=t+e^{-t}\) (A) \(\frac{e^{-1}}{1-e^{-t}}\) (B) \(e^{-t}+1\) (C) \(e^{t}-e^{-2 t}\) (D) \(e^{t}-1\)