Section I Part B — Full Practice Exam 2

Questions in this quiz (15)

1. Let $$f$$ be a function such that $$f(1)=2$$ and $$f(4)=8$$. Which of the following conditions guarantees that there exists a value $$c$$, $$1$$<$$c$$<$$4$$, for which $$f^{\prime}(c)=2$$?

   • $$f$$ is continuous on $$(1,4)$$
   • $$f$$ is differentiable on $$(1,4)$$ and continuous on $$[1,4]$$
   • $$f$$ is increasing on $$[1,4]$$
   • $$f^{\prime}(x)$$ exists at one point in $$(1,4)$$

2. The rate at which water enters a tank is modeled by$$R(t)=2+\cos(t^2)$$ for $$t\ge0$$, where $$R(t)$$ is measured in liters per minute and $$t$$ is measured in minutes. How much water enters the tank during the first $$0.5$$ minutes?

   • $$1.006$$ liters
   • $$1.248$$ liters
   • $$0.812$$ liters
   • $$0.504$$ liters

3. The function $$f$$ has derivatives of all orders with $$f(0)=1$$, $$f^{\prime}(0)=2$$, $$f^{\prime\prime}(0)=-3$$, $$f^{\prime\prime\prime}(0)=6.$$ Let $$g(x)=\int_0^x f(t)\,dt.$$ What is the third-degree Taylor polynomial for $$g$$ about $$x=0$$?

   • $$x+x^2-x^3$$
   • $$x+x^2-\frac{1}{2}x^3$$
   • $$x+x^2-x^3+\frac{1}{4}x^4$$
   • $$x+2x^2-x^3$$

4. If $$f(x)=(x+1)\sin(\sqrt{x+1}),$$ what is the average value of $$f$$ on the interval $$[0,3]$$?

   • $$1.842$$
   • $$2.409$$
   • $$3.205$$
   • $$4.118$$

5. The infinite series $$\sum_{k=1}^{\infty} a_k$$ has nth partial sum$$S_n=\frac{2^n}{4^{n+1}}.$$ What is the sum of the series?

   • $$0$$
   • $$\frac{2}{3}$$
   • $$1$$
   • The series diverges

6. Let $$y=f(x)$$ be twice differentiable and let $$y=t(x)$$ be the tangent line to the graph at $$x=1$$. If $$t(x)\le f(x)$$ for all real $$x$$, which must be true?

   • $$f^{\prime}(1)\ge0$$
   • $$f^{\prime\prime}(1)\ge0$$
   • $$f^{\prime\prime}(1)\le0$$
   • $$f(1)\ge0$$

7. Let $$f$$ be twice differentiable with values shown:What is the value of $$\int_0^1 x f^{\prime\prime}(x)\,dx?$$

   • $$2$$
   • $$3$$
   • $$-1$$
   • $$-4$$

8. The first derivative of a function is given by$$f^{\prime}(x)=\cos(x^2).$$ At which value of $$x$$ does $$f$$ have a local minimum?

   • $$0.886$$
   • $$1.253$$
   • $$1.772$$
   • $$2.171$$

9. The function $$f(x)=2x-3\cos(3x).$$ What value satisfies the Mean Value Theorem on $$[0,2]$$?

   • $$0.412$$
   • $$0.729$$
   • $$1.045$$
   • $$1.587$$

10. The shaded region in the figure above is bounded by the graph of $$y=\sqrt{\cos\left(\frac{\pi x}{10}\right)}$$ and the lines $$x=-7,\ x=7,\ y=2.$$ What is the area of this region?

    • $$6.372$$
    • $$7.628$$
    • $$20.372$$
    • $$24.923$$

11. The figure above shows the graph of $$f'$$, the derivative of a function $$f$$, for $$0\le x\le2$$. What is the value of $$x$$ at which the absolute minimum of $$f$$ occurs?

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

12. Water is poured into an inverted cone with height $$20\ ft$$ and radius $$5\ ft$$ at a rate of $$12$$ cubic feet per minute. How fast is the water level rising when the water is $$8\ ft$$ deep?

    • $$0.191\ ft/min$$
    • $$0.382\ ft/min$$
    • $$0.955\ ft/min$$
    • $$1.146\ ft/min$$

13. The graphs of the differentiable functions $$f$$ and $$g$$ are shown above. If the function $$p$$ is defined by$$p\left(x\right)=f\left(x\right) \cdot g\left(x\right)$$which of the following must be true about $$p'$$, the derivative of $$p$$?

    • $$p'\left(-2\right)$$<$$0$$
    • $$p'\left(-2\right)=0$$
    • $$p'\left(-2\right)$$>$$0$$
    • $$p'\left(0\right)$$<$$0$$

14. The graph of $$f'$$, the derivative of the function $$f$$, is shown in the figure above. Which of the following statements about $$f$$ at $$x=-2$$ is true?

    • $$f$$ is not continuous at $$x=-2$$
    • The graph of $$f$$ has a vertical tangent line at $$x=-2$$
    • The graph of $$f$$ has a point of inflection at $$x=-2$$
    • The derivative of $$f$$ does not exist at $$x=-2$$

15. The alternating series test can be used to show convergence of which of the following alternating series?$$I.$$ $$3-\frac{1}{4}+\frac{1}{9}-\frac{1}{16}+\frac{1}{25}-\cdots+a_n+\cdots,$$ where$$a_n=\left(-1\right)^{n+1}\frac{1}{n^2}$$$$II.$$ $$2-\frac{2}{3}+\frac{2}{5}-\frac{2}{7}+\frac{2}{9}-\cdots+a_n+\cdots,$$ where$$a_n=\left(-1\right)^{n+1}\frac{2}{2n-1}$$$$III.$$ $$\frac{3}{2}-\frac{4}{3}+\frac{5}{4}-\frac{6}{5}+\frac{7}{6}-\cdots+a_n+\cdots,$$ where$$a_n=\left(-1\right)^{n+1}\frac{n+2}{n+1}$$

    • $$I$$ only
    • $$II$$ only
    • $$I$$ and $$II$$ only
    • $$I,\ II,\ and\ III$$