Mini Exam 2 — Timed Mini Exams

Questions in this quiz (10)

1. Let $$f$$ be defined as $$f(x)= x^2+kx$$ & $$x$$<$$1$$ $$f(x)=\ln(x+1)$$, $$x\ge1$$ for some constant $$k$$. For what value of $$k$$ will $$f$$ be differentiable over its entire domain?

   • $$-1$$
   • $$0$$
   • $$1$$
   • No value

2. What is the approximation of the value of $$e^2$$ obtained by using a third-degree Taylor polynomial about $$x=0$$ for $$e^x$$?

   • $$1+2+2$$
   • $$1+2+2+\frac{4}{3}$$
   • $$1+2+\frac{4}{3}$$
   • $$2+2+\frac{4}{3}$$

3. Consider the region in the first quadrant bounded by $$y=x^3$$ over $$[0,2]$$. Let $$L_2$$ represent the left Riemann sum, $$R_2$$ represent the right Riemann sum, $$M_2$$ represent the midpoint approximation, and $$T_2$$ represent the trapezoidal approximation. Which statement is true?

   • $$L_2
   • $$R_2<T_2<M_2<L_2$$
   • $$M_2<L_2<R_2<T_2$$
   • $$L_2<T_2<M_2<R_2$$

4. Evaluate $$\int 4x^2 e^{2x}\,dx$$

   • $$e^{2x}(2x^2-2x+1)+C$$
   • $$e^{2x}(x^2-2x+2)+C$$
   • $$e^{2x}(2x^2-x+2)+C$$
   • $$e^{2x}(x^2-x+1)+C$$

5. Which of the following series converge? I. $$\sum_{n=1}^{\infty}\frac{3^n}{n^2}$$ II. $$\sum_{n=1}^{\infty}\frac{1}{n^{3/2}}$$ III. $$\sum_{n=1}^{\infty}\frac{\sin n}{n}$$

   • I only
   • II only
   • II and III
   • I and III

6. When $$x=9$$, the rate at which $$x^{5/2}$$ is increasing is $$k$$ times the rate at which $$\sqrt{x}$$ is increasing. What is the value of $$k$$?

   • $$\frac{3}{2}$$
   • $$3$$
   • $$\frac{15}{2}$$
   • $$405$$

7. The area of the region inside the polar curve $$r=3\sin\theta$$ but outside the polar curve $$r=1$$ is given by

   • $$\frac{1}{2}\int_{\pi/6}^{5\pi/6}(9\sin^2\theta-1)\,d\theta$$
   • $$\int_{\pi/6}^{5\pi/6}(3\sin\theta-1)\,d\theta$$
   • $$\frac{1}{2}\int_0^\pi(9\sin^2\theta-1)\,d\theta$$
   • $$\frac{1}{2}\int_{\pi/6}^{5\pi/6}(3\sin\theta-1)^2\,d\theta$$

8. The length of the path described by the parametric equations $$x=\sin2t,\ y=\cos t$$ for $$0\le t\le\pi$$ is given by

   • $$\int_0^\pi\sqrt{4\cos^22t+\sin^2t}\,dt$$
   • $$\int_0^\pi\sqrt{4\cos^22t+\sin^2t\cos^2t}\,dt$$
   • $$\int_0^\pi\sqrt{2\sin^22t+\cos^2t}\,dt$$
   • $$\int_0^\pi\sqrt{4\cos^22t+\sin^2t}\,dt$$

9. Determine the interval of convergence for the series $$\sum_{n=0}^{\infty}\frac{(2x-1)^{n+1}}{n^{3/2}}$$

   • $$-\frac{1}{2}\le x\le\frac{1}{2}$$
   • $$0\le x\le1$$
   • $$-\frac{1}{2}\le x<1$$
   • $$\frac{1}{2}\le x\le1$$

10. $$f(x)=\frac{(2x+1)(x-3)}{(x-1)(x+2)}$$ has a horizontal asymptote at $$x=$$

    • $$0$$
    • $$1$$
    • $$2$$
    • None of the above