Section I Part A — Full Practice Exam 1

Questions in this quiz (30)

1. What is the instantaneous rate of change for the function $$g(x)=\frac{x^{3}-3x^{2}+x-2}{x-1},$$ when $$x=2$$?

   • $$-5$$
   • $$-1$$
   • $$1$$
   • $$5$$

2. The graph shows the rate in bikes per minute at which bicycles pass through a park entrance over a 24 hour period. Using the graph, the total number of bicycles that pass through the entrance by 3:00 PM can be best estimated as:

   • $$45$$
   • $$720$$
   • $$1080$$
   • $$27000$$

3. Evaluate the definite integral:$$\int_{2}^{5}\frac{3x}{x^4}dx$$

   • $$\frac{63}{200}$$
   • $$\frac{9}{20}$$
   • $$\frac{27}{40}$$
   • $$\frac{3}{4}$$

4. The table below gives values of a function $$g,$$ which is continuous on the closed interval $$[1,3].$$Which of the following statements must be true?

   • $$g$$ has no inflection points on $$[1,3]$$
   • The average rate of change of $$g$$ on $$[1,3]$$ is positive
   • $$g'(2)=4$$
   • $$g$$ is linear on $$[1,3]$$

5. Evaluate the definite integral:$$\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{\sec x\tan x}{\cos x}dx$$

   • $$e-1$$
   • $$1-e$$
   • $$e$$
   • This integral does not have a finite value.

6. At the point $$(2,1)$$, what is the slope of the curve defined implicitly by:$$x^2+4y^2+2xy-6x-8y=0?$$

   • $$-\frac{3}{2}$$
   • $$-\frac{1}{2}$$
   • $$0$$
   • It is undefined

7. Evaluate the integral:$$\int_0^1\sqrt{x}(3x^2-2x+1)dx$$

   • $$\frac{8}{15}$$
   • $$\frac{76}{105}$$
   • $$\frac{24}{35}$$
   • $$\frac{32}{35}$$

8. The radius of a sphere is increasing at a rate of $$2$$ inches per minute. When the surface area of the sphere is $$16\pi$$ square inches, how fast is the volume increasing in cubic inches per minute?

   • $$8\pi$$
   • $$16\pi$$
   • $$32\pi$$
   • $$64\pi$$

9. Using the graph shown, the shaded region is bounded by the curves $$y=x^3$$ and $$y=2x+4$$. Which of the following integrals correctly represents the area of the shaded region?

   • $$\int_0^2(2x+4-x^3)dx$$
   • $$\int_2^0(2x+4-x^3)dx$$
   • $$\int_0^2(x^3-2x-4)dx$$
   • $$\int_0^8(2x+4-x^3)dx$$

10. What is the average rate of change of $$g(x)=x^3-2x^2+5x-1$$ over the interval $$[1,4]$$?

    • $$6$$
    • $$9$$
    • $$12$$
    • $$16$$

11. If the graph of the second derivative of a function $$f$$ is a line with nonzero slope, what type of function could $$f$$ be?

    • Linear
    • Quadratic
    • Cubic
    • Quartic

12. Let the function $$f$$ be defined by$$f(x)=\sqrt{4-x}$$ for $$x<1$$ $$f(x)=x^2$$ for $$x\ge1$$What is the average value of $$f$$ on the interval $$[0,3]$$?

    • $$\frac{7}{3}$$
    • $$\frac{8}{3}$$
    • $$\frac{1}{3}\left(14-2\sqrt{3}\right)$$
    • $$\frac{10}{3}$$

13. The graph shown represents a function $$g$$ that is twice differentiable. At $$x=2$$, the graph has a horizontal tangent and is concave up. Which of the following statements must be true?

    • $$g''(2)<g'(2)<g(2)$$
    • $$g'(2)<g(2)<g''(2)$$
    • $$g(2)<g'(2)<g''(2)$$
    • $$g(2)

14. Let $$f$$ be continuous on $$[-1,5]$$ and differentiable on $$(-1,5)$$. If $$f(-1)=2$$ and $$f(5)=8$$, which statement is guaranteed by the Mean Value Theorem?

    • $$f'(2)=1$$
    • $$f'(x)=1$$ for all $$x\in(-1,5)$$
    • There exists some $$c\in(-1,5)$$ such that $$f'(c)=1$$
    • $$f$$ is increasing on $$[-1,5]$$

15. Evaluate:$$\frac{d}{dx}\left(\int_{0}^{x^2}e^tdt\right)$$

    • $$e^{x^2}$$
    • $$2xe^{x^2}$$
    • $$x^2e^x$$
    • $$e^x$$

16. Assume $$f(x)=e^x.$$ If the rate of change of $$f$$ at $$x=c$$ is $$e^3$$ times the rate of change at $$x=1,$$ then $$c=$$?

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

17. The table below gives values of $$f,$$ $$g,$$ and their derivatives.Let $$h(x)=f(g(x)).$$ For which value of $$x$$ is $$h'(x)=0$$?

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

18. Let $$f$$ be differentiable on $$[0,6]$$ with $$f(0)=2$$ and $$f(6)=2$$. If the equation $$f(x)=5$$ has exactly two solutions in $$(0,6),$$ which statement must be true?

    • $$f$$ is increasing on $$(0,6)$$
    • $$f'(x)=0$$ for some $$x\in(0,6)$$
    • $$f$$ has an absolute minimum at $$x=3$$
    • $$f(x)\ge 2$$ for all $$x\in[0,6]$$

19. Find the slope of the normal line to the curve $$y=\sqrt{x+1}$$ at the point $$(3,2).$$

    • $$-4$$
    • $$-2$$
    • $$-\frac{1}{2}$$
    • $$-\frac{1}{4}$$

20. For which values of $$k$$ does $$\int_{-2}^{k}x^3dx=0$$?

    • $$0$$
    • $$2$$
    • $$-2$$ and $$2$$
    • $$-2$$, $$0$$ and $$2$$

21. If a quantity $$y$$ is directly proportional to its rate of change, which of the following could represent $$y$$ as a function of $$t$$?

    • $$y=5e^{3t}$$
    • $$y=t^4$$
    • $$y=\ln t$$
    • $$y=\frac{1}{t}$$

22. On which interval(s) is the graph of $$y=3x^3-6x^2+2$$ decreasing?

    • $$(-\infty,0)$$
    • $$(0,\frac{4}{3})
    • $$(1,\infty)$$
    • $$(0,1)\cup(1,\infty)$$

23. For $$f(x)=\frac{x^2+4}{x}$$, find the value of $$c$$ in $$[1,4]$$ that satisfies the Mean Value Theorem.

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

24. A slope field for the differential equation $$\frac{dy}{dx}=g(x)$$ shows slopes that are zero at $$x=0$$, positive for $$x>0$$, and negative for $$x<0$$. Which function could represent $$g(x)$$?

    • $$g(x)=x$$
    • $$g(x)=x^2$$
    • $$g(x)=|x|$$
    • $$g(x)=-1$$

25. A slope field for the differential equation $$\frac{dy}{dx}=g(x,y)$$ is shown. Based on the features of the slope field, which of the following differential equations is the simplest that could generate the displayed pattern?

    • $$\frac{dy}{dx}=x+y$$
    • $$\frac{dy}{dx}=xy$$
    • $$\frac{dy}{dx}=x-y$$
    • $$\frac{dy}{dx}=x^2+y^2$$

26. In the diagram above, the curves $$y=g(x)$$ and $$y=f(x)$$ intersect at $$x=a$$ and $$x=b$$, with $$g(x)\ge f(x)$$ on $$[a,b]$$. Which expression represents the net signed area between the two curves over the interval $$[a,b]$$?

    • $$\int_a^b(g(x)-f(x))dx$$
    • $$\int_b^a(f(x)-g(x))dx$$
    • $$\int_a^b(f(x)-g(x))dx$$
    • $$\int_b^a(g(x)-f(x))dx$$

27. On the closed interval $$[2,6]$$, the function $$f$$ is continuous. Suppose $$f(2)=4$$ and $$f(6)=1$$. If $$f'(x)<0$$ and $$f''(x)>0$$ for all $$x\in(2,6)$$, which of the following could be the value of $$f(4)$$?

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

28. The water level in a cylindrical tank is decreasing at a rate of $$3$$ inches per minute. The radius of the tank is $$18$$ inches. What is the rate at which the volume of water is changing, in cubic inches per minute, when the volume is $$324\pi$$ cubic inches?

    • $$972\pi$$
    • $$-972\pi$$
    • $$-486\pi$$
    • $$486\pi$$

29. If $$f(x)=\arctan(x^2)$$, find $$f'(1).$$

    • $$\frac{1}{2}$$
    • $$1$$
    • $$\frac{2}{5}$$
    • $$\frac{1}{5}$$

30. Beginning at the origin, a particle moves along the positive x-axis with a velocity that remains positive but steadily decreases over time. Which of the following graphs could represent the particle's position $$s(t)$$ as a function of time $$t$$?

    • A straight line through the origin with constant positive slope
    • A curve starting at the origin that is increasing and concave down
    • A curve starting above the origin that is increasing and concave up
    • A curve starting at the origin that decreases for some time before increasing