Mini Exam 7 — Timed Mini Exams

Questions in this quiz (10)

1. Evaluate $$\int_0^4\frac{e^{\sqrt{x}}}{\sqrt{x}}\,dx$$

   • $$2(e^2-1)$$
   • $$2(e^2-1)$$
   • $$e^2-1$$
   • $$e^2-1$$
   • $$4(e^2-1)$$
   • $$4(e^2-1)$$
   • $$2(e^4-1)$$
   • $$2(e^4-1)$$

2. The length of the path described by the parametric equations $$x=\frac{1}{2}t^2,\quad y=\frac{1}{3}t^3,\quad 0\le t\le3$$ is

   • $$\int_0^3\sqrt{t^2+t^4}\,dt$$
   • $$\int_0^3\sqrt{t^2+t^4}\,dt$$
   • $$\int_0^3\sqrt{t^2+\frac{4}{9}t^4}\,dt$$
   • $$\int_0^3\sqrt{t^2+\frac{4}{9}t^4}\,dt$$
   • $$\int_0^3\sqrt{\frac{1}{4}t^2+t^4}\,dt$$
   • $$\int_0^3\sqrt{\frac{1}{4}t^2+t^4}\,dt$$
   • $$\int_0^3\sqrt{t^2+\frac{1}{9}t^4}\,dt$$
   • $$\int_0^3\sqrt{t^2+\frac{1}{9}t^4}\,dt$$

3. For what integer $$k>1$$ will both $$\sum_{n=1}^{\infty}\frac{(-1)^n}{n}$$ and $$\sum_{n=1}^{\infty}\left(\frac{k}{4}\right)^n$$ converge?

   • $$2$$
   • $$2$$
   • $$3$$
   • $$3$$
   • $$4$$
   • $$4$$
   • $$5$$
   • $$5$$

4. The volume of the solid formed when the region bounded by $$y=\sqrt{9-x^2},\quad x=0,\quad y=0$$ is rotated about the line $$y=-3$$ is given by which definite integral?

   • $$\pi\int_0^3(9-x^2)\,dx$$
   • $$\pi\int_0^3(9-x^2)\,dx$$
   • $$\pi\int_0^3\left[\left(\sqrt{9-x^2}+3\right)^2-9\right]\,dx$$
   • $$\pi\int_0^3\left[\left(\sqrt{9-x^2}+3\right)^2-9\right]\,dx$$
   • $$2\pi\int_0^3\sqrt{9-x^2}\,dx$$
   • $$2\pi\int_0^3\sqrt{9-x^2}\,dx$$
   • $$\pi\int_0^3(\sqrt{9-x^2})^2\,dx$$
   • $$\pi\int_0^3(\sqrt{9-x^2})^2\,dx$$

5. If $$f$$ is a vector-valued function defined by $$f(t)=\langle e^{3t},\sin2t\rangle$$ then $$f''(t)=$$

   • $$\langle9e^{3t},-4\sin2t\rangle$$
   • $$\langle9e^{3t},-4\sin2t\rangle$$
   • $$\langle3e^{3t},2\cos2t\rangle$$
   • $$\langle3e^{3t},2\cos2t\rangle$$
   • $$\langle9e^{3t},4\cos2t\rangle$$
   • $$\langle9e^{3t},4\cos2t\rangle$$
   • $$\langle9e^{3t},-4\cos2t\rangle$$
   • $$\langle9e^{3t},-4\cos2t\rangle$$

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

   • $$\frac{e^{2x}}{5}(2\cos x+\sin x)+C$$
   • $$\frac{e^{2x}}{5}(2\cos x+\sin x)+C$$
   • $$\frac{e^{2x}}{5}(2\cos x-\sin x)+C$$
   • $$\frac{e^{2x}}{5}(2\cos x-\sin x)+C$$
   • $$e^{2x}(2\cos x+\sin x)+C$$
   • $$e^{2x}(2\cos x+\sin x)+C$$
   • $$e^{2x}(\cos x-\sin x)+C$$
   • $$e^{2x}(\cos x-\sin x)+C$$

7. The graph of the function represented by the Maclaurin series $$1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots$$ intersects the graph of $$y=2x^3-x+2$$ at $$x=$$

   • $$-1.215$$
   • $$-1.104$$
   • $$-1.050$$
   • $$-0.985$$

8. The acceleration of a particle is described by the parametric equations $$x''(t)=2t+1,\quad y''(t)=\frac{2}{t}$$ If the velocity vector of the particle when $$t=2$$ is $$\langle5,2\ln2\rangle$$, what is the velocity vector of the particle when $$t=1$$?

   • $$\langle3,\ln2\rangle$$
   • $$\langle4,\ln2\rangle$$
   • $$\langle3,2\ln2\rangle$$
   • $$\langle4,2\ln2\rangle$$

9. Evaluate $$\int\frac{3x^2+2x+1}{x^2+x-2}\,dx$$

   • $$3x+5\ln|x+2|-\ln|x-1|+C$$
   • $$3x+5\ln|x+2|+\ln|x-1|+C$$
   • $$3x+\ln|x+2|-\ln|x-1|+C$$
   • $$3x+5\ln|(x+2)(x-1)|+C$$

10. The revenue from the sale of a product is $$R(x)=120x+800$$ dollars, and the total production cost is $$C(x)=2x^2+20x+200$$ dollars, where $$x$$ is the number of items produced. How many items should be made to maximize profit?

    • $$5$$
    • $$10$$
    • $$15$$
    • $$20$$