1.
Question 1. \(\lim_{x\to 0}\frac{\sin(4x)}{\sin(5x)}=\)
Use the standard limit \(\lim_{u\to 0}\frac{\sin u}{u}=1\). Rewrite the expression as \(\frac{\sin(4x)}{\sin(5x)}=\frac{4}{5}\cdot\frac{\sin(4x)}{4x}\cdot\frac{5x}{\sin(5x)}\). As \(x\to0\), both \(\frac{\sin(4x)}{4x}\) and \(\frac{5x}{\sin(5x)}\) approach \(1\). Therefore, the limit is \(\frac{4}{5}\). Answer: A
2.
Question 2. \(g(x)=\frac{x+3}{x-2}\) has a removable discontinuity at \(x=2\).
A removable discontinuity happens when a factor in the denominator cancels with a matching factor in the numerator. For \(g(x)=\frac{x+3}{x-2}\), the factor \(x-2\) does not cancel. Therefore, the discontinuity at \(x=2\) is not removable. Answer: A
3.
Question 3. \(f(x)=x^5-3x^2+x-7\) is continuous everywhere.
The function \(f(x)=x^5-3x^2+x-7\) is a polynomial. All polynomial functions are continuous for all real numbers. Therefore, \(f(x)\) is continuous everywhere. Answer: A
4.
Question 4. \(\lim_{x\to\infty}\frac{\cos(3x)}{x}=\)
The numerator \(\cos(3x)\) is bounded between \(-1\) and \(1\), while the denominator \(x\) grows without bound as \(x\to\infty\). A bounded quantity divided by a number growing to infinity approaches \(0\). Therefore, the limit is \(0\). Answer: B
5.
Question 5. The average rate of change of \(y\) with respect to \(x\) over the interval \([0,3]\) for the function \(y=3x+2\) is \(9\).
The average rate of change is \(\frac{y_2-y_1}{x_2-x_1}\). For \(y=3x+2\), \(y(0)=3(0)+2=2\), and \(y(3)=3(3)+2=11\). Thus the average rate of change is \(\frac{11-2}{3-0}=\frac{9}{3}=3\), not \(9\). Answer: B
6.
Question 6. What is the instantaneous rate of change of \(y=\sqrt{x}\) at \(x=2\)?
The instantaneous rate of change is the derivative. Since \(y=\sqrt{x}=x^{1/2}\), we have \(y'=\frac{1}{2}x^{-1/2}=\frac{1}{2\sqrt{x}}\). At \(x=2\), \(y'=\frac{1}{2\sqrt{2}}\). Answer: C
7.
Question 7. The slope of the tangent to \(f(x)=3x^2-1\) at \(x=2\) is \(12\).
The slope of the tangent is the derivative. For \(f(x)=3x^2-1\), \(f'(x)=6x\). At \(x=2\), \(f'(2)=6(2)=12\). Therefore, the statement is true. Answer: A
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