Mini Exam 5 — Timed Mini Exams

Questions in this quiz (10)

1. The rational function f is given by $$f(x)=\frac{x^3+2x^2-5x}{x^2-x}$$. In the $$xy$$-plane, which of the following is the slope of a slant asymptote of the graph of $$f$$?

   • $$0$$
   • $$1$$
   • $$2$$
   • $$3$$

2. Let $$f(x)=\cos x$$. The function $$g$$ has the same amplitude as $$f$$, and the period of $$g$$ is $$\frac{1}{3}$$ of the period of $$f$$. Which of the following defines $$g$$ in terms of $$f$$?

   • $$g(x)=f\left(\frac{1}{3}x\right)$$
   • $$g(x)=3f(x)$$
   • $$g(x)=f(3x)$$
   • $$g(x)=\frac{1}{3}f(x)$$

3. The binomial theorem can be used to expand an expression of the form $$(a+b)^n$$. Which of the following is equivalent to $$(x+2y)^5$$?

   • $$x^5+(2y)^5$$
   • $$x^5+x^4(2y)+x^3(2y)^2+x^2(2y)^3+x(2y)^4+(2y)^5$$
   • $$x^5+5x^4(2y)+10x^3(2y)^2+10x^2(2y)^3+5x(2y)^4+(2y)^5$$
   • $$x^5+4x^4(2y)+6x^3(2y)^2+4x^2(2y)^3+(2y)^5$$

4. The polar function $$r=f(\theta)$$, where $$f(\theta)=2-3\cos(-\theta)$$, is graphed in the polar coordinate system. As $$\theta$$ varies from $$\frac{\pi}{2}$$ to $$\pi$$, how is the distance between the origin and the point $$(f(\theta),\theta)$$ changing?

   • The distance remains constant.
   • The distance is decreasing.
   • The distance is increasing.
   • The distance decreases, then increases.

5. The function $$f$$ is given by $$f(x)=x-2$$, and the function $$g$$ is given by $$g(x)=(x-2)(2x+1)$$. Consider the rational function $$m(x)=\frac{f(x)}{g(x)}$$. Which of the following is true about holes in the graph of $$m$$ in the $$xy$$-plane?

   • The graph of $$m$$ has no holes.
   • The graph of $$m$$ has holes at $$x=-\frac{1}{2}$$ and $$x=2$$.
   • The graph of $$m$$ has a hole at $$x=-\frac{1}{2}$$.
   • The graph of $$m$$ has a hole at $$x=2$$.

6. Consider the functions $$g$$ and $$h$$ given by $$g(\theta)=\tan\theta$$ and $$h(\theta)=\sec\theta$$. Which of the following statements is true for both $$g$$ and $$h$$?

   • The domain is all real numbers.
   • The range is $$(-\infty,-1]\cup[1,\infty)$$.
   • The graphs have vertical asymptotes at $$\theta=\frac{\pi}{2}+\pi n$$, where $$n$$ is an integer.
   • The graphs have vertical asymptotes at $$\theta=\pi n$$, where $$n$$ is an integer.

7. Both nonzero functions $$f$$ and $$g$$ are invertible. The input values of $$f$$ are distances, in miles, and the outputs are times, in hours. The input values of $$g$$ are times, in hours, and the outputs are costs, in dollars. For which of the following are the input values costs, in dollars, and the output values distance, in miles?

   • $$y=g(f(x))$$
   • $$y=f^{-1}(g^{-1}(x))$$
   • $$y=g^{-1}(f^{-1}(x))$$
   • $$y=f(g(x))$$

8. A data set is modeled using both linear and exponential regressions. The residual plot for the linear model shows a clear curved pattern, while the residual plot for the exponential model shows points randomly scattered around zero. Which of the following statements is true?

   • A linear model is more appropriate, and the curved residual plot corresponds to the linear model.
   • A linear model is more appropriate, and the random residual plot corresponds to the linear model.
   • An exponential model is more appropriate, and the curved residual plot corresponds to the exponential model.
   • An exponential model is more appropriate, and the random residual plot corresponds to the exponential model.

9. Which of the following values is a zero of the function $$h(x)=2\cos(x+1.2)-\sin(x-0.4)$$ on the interval $$[0,\pi]$$?

   • $$0.381$$
   • $$1.674$$
   • $$2.103$$
   • $$2.765$$

10. In the $$xy$$-plane, the function $$f$$ is graphed and contains the point $$(4,3)$$. Which of the following points is on the graph of $$y=3f(x)-2$$?

    • $$(4,3)$$
    • $$(4,7)$$
    • $$(3,4)$$
    • $$(6,3)$$