Practise Test

Part-A

• 1. If \( f(x) = (2x^2 + 5)^7 \), then \( f'(x) = \)

  \( 7(4x)^6 \)
  \( 7(2x^2 + 5)^6 \)
  \( 14x^2(2x^2 + 5)^6 \)
  \( 28x(2x^2 + 5)^6 \)
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• 2. \( \quad \displaystyle \int \frac{1}{3x + 12} \, dx = \)

  \( -3 \ln |x + 4| + C \)
  \( \frac{1}{3} \ln |x + 4| + C \)
  \( \ln |x + 4| + C \)
  \( 3 \ln |x + 4| + C \)
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• 3. \( \quad \text{If } f(x) = \dfrac{5 - x}{x^3 + 2}, \text{ then } f'(x) = \)

  \( \dfrac{-4x^3 + 15x^2 - 2}{(x^3 + 2)^2} \)
  \( \dfrac{-2x^3 + 15x^2 + 2}{(x^3 + 2)^2} \)
  \( \dfrac{2x^3 - 15x^2 - 2}{(x^3 + 2)^2} \)
  \( \dfrac{4x^3 - 15x^2 + 2}{(x^3 + 2)^2} \)
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• 4. The table above gives the velocity \( v(t) \), in miles per hour, of a truck at selected times \( t \), in hours. Using a trapezoidal sum with the three subintervals indicated by the table, what is the approximate distance, in miles, the truck traveled from \( t = 0 \) to \( t = 3 \)?

  \( t \)	0	0.5	2	3
  \( v(t) \)	20	60	40	30

  140
  130
  125
  120
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• 5. If \( f(x)=\sin(x^{2}+\pi) \), then \( f'(\sqrt{2\pi}) = \)

  \( -2\sqrt{2\pi} \)
  \( -2 \)
  \( -1 \)
  \( \cos(2\sqrt{2\pi}) \)
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• 6. If \( f \) is the function given by \( f(x)=3x^{2}-x^{3} \), then the average rate of change of \( f \) on the closed interval \([1,5]\) is

  \( -21 \)
  \( -13 \)
  \( -12 \)
  \( -9 \)
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• 7. If \[ \int_{4}^{-10} g(x)\,dx = -3 \quad \text{and} \quad \int_{4}^{6} g(x)\,dx = 5, \] then \[ \int_{-10}^{6} g(x)\,dx = \]

  \( -8 \)
  \( -2 \)
  \( 2 \)
  \( 8 \)
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• 8. If \( f \) is the function given by \( f(x) = e^{x/3} \), which of the following is an equation of the line tangent to the graph of \( f \) at the point \( (3\ln 4, 4) \)?

  \( y - 4 = \dfrac{4}{3}(x - 3\ln 4) \)
  \( y - 4 = 4(x - 3\ln 4) \)
  \( y - 4 = 12(x - 3\ln 4) \)
  \( y - 3\ln 4 = 4(x - 4) \)
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• 9. The graph of a function \( f \) is shown above. Which of the following expresses the relationship between \( \displaystyle \int_{0}^{2} f(x)\,dx \), \( \displaystyle \int_{0}^{3} f(x)\,dx \), and \( \displaystyle \int_{2}^{3} f(x)\,dx \) ?

  
  \( \displaystyle \int_{0}^{2} f(x)\,dx < \int_{0}^{3} f(x)\,dx < \int_{2}^{3} f(x)\,dx \)
  \( \displaystyle \int_{0}^{3} f(x)\,dx < \int_{0}^{2} f(x)\,dx < \int_{2}^{3} f(x)\,dx \)
  \( \displaystyle \int_{2}^{3} f(x)\,dx < \int_{0}^{2} f(x)\,dx < \int_{0}^{3} f(x)\,dx \)
  \( \displaystyle \int_{2}^{3} f(x)\,dx < \int_{0}^{3} f(x)\,dx < \int_{0}^{2} f(x)\,dx \)
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• 10. \( \displaystyle \int_{0}^{2} (x^{3} + 1)^{1/2} x^{2} \, dx = \)

  \( \dfrac{52}{9} \)
  \( 6 \)
  \( \dfrac{26}{3} \)
  \( \dfrac{52}{3} \)
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• 11. If \( x^{2} + xy - 3y = 3 \), then at the point \( (2, 1) \), \( \dfrac{dy}{dx} = \)

  \( 5 \)
  \( 4 \)
  \( \dfrac{7}{3} \)
  \( 2 \)
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• 12. The number of gallons of water in a storage tank at time \( t \), in minutes, is modeled by \( w(t) = 25 - t^{2} \) for \( 0 \le t \le 5 \). At what rate, in gallons per minute, is the amount of water in the tank changing at time \( t = 3 \) minutes?

  \( 66 \)
  \( 16 \)
  \( -3 \)
  \( -6 \)
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• 13. Shown above is a slope field for which of the following differential equations?

  
  \( \dfrac{dy}{dx} = \dfrac{x(4 - y)}{4} \)
  \( \dfrac{dy}{dx} = \dfrac{y(4 - y)}{4} \)
  \( \dfrac{dy}{dx} = \dfrac{xy(4 - y)}{4} \)
  \( \dfrac{dy}{dx} = \dfrac{y^{2}(4 - y)}{4} \)
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• 14. The weight of a population of yeast is given by a differentiable function \( y \), where \( y(t) \) is measured in grams and \( t \) is measured in days. The weight of the yeast population increases according to the equation \( \dfrac{dy}{dt} = ky \), where \( k \) is a constant. At time \( t = 0 \), the weight of the yeast population is 120 grams and is increasing at the rate of 24 grams per day. Which of the following is an expression for \( y(t) \)?

  \( 120e^{24t} \)
  \( 120e^{t/5} \)
  \( e^{t/5} + 119 \)
  \( 24t + 120 \)
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• 15. The graphs of the functions \( f \) and \( g \) are shown in the figures above. Which of the following statements is false?

  
  \( \displaystyle \lim_{x \to 1} f(x) = 0 \)
  \( \displaystyle \lim_{x \to 2} g(x) \) does not exist.
  \( \displaystyle \lim_{x \to 1} \big( f(x)g(x + 1) \big) \) does not exist.
  \( \displaystyle \lim_{x \to 1} \big( f(x + 1)g(x) \big) \) exists.
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• 16. Let \(f\) be the function defined by \(f(x)=-3+6x^2-2x^3\). What is the largest open interval on which the graph of \(f\) is both concave up and increasing?

  \( (0,1) \)
  \( (1,2) \)
  \( (0,2) \)
  \( (2,\infty) \)
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• 17. A particle moves along the \(x\)-axis so that at time \(t > 0\) its position is given by \(x(t) = 12e^{-t} \sin t\). What is the first time \(t\) at which the velocity of the particle is zero?

  \( \frac{\pi}{4} \)
  \( \frac{\pi}{2} \)
  \( \frac{3\pi}{4} \)
  \( \pi \)
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• 18. Let \( F \) be the function given by \( F(x) = \displaystyle \int_{3}^{x} \left( \tan(5t)\sec(5t) - 1 \right) dt \). Which of the following is an expression for \( F'(x) \)?

  \( \frac{1}{5}\sec(5x) - 1 \)
  \( \frac{1}{5}\sec(5x) - x \)
  \( \tan(5x)\sec(5x) \)
  \( \tan(5x)\sec(5x) - 1 \)
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• 19. Let \( f \) be the function given by \( f(x) = 2\cos x + 1 \). What is the approximation for \( f(1.5) \) found by using the line tangent to the graph of \( f \) at \( x = \frac{\pi}{2} \)?

  \( -2 \)
  \( 1 \)
  \( \pi - 2 \)
  \( 4 - \pi \)
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• 20. Let \( f \) be the function given by \( f(x) = \dfrac{x - 2}{2|x - 2|} \). Which of the following is true?

  \( \displaystyle \lim_{x \to 2} f(x) = \frac{1}{2} \)
  \( f \) has a removable discontinuity at \( x = 2 \).
  \( f \) has a jump discontinuity at \( x = 2 \).
  \( f \) has a discontinuity due to a vertical asymptote at \( x = 2 \).
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• 21. If \( f(x) = \ln x \), then \( \displaystyle \lim_{x \to 3} \frac{f(x) - f(3)}{x - 3} \) is

  \( \frac{1}{3} \)
  \( e^3 \)
  \( \ln 3 \)
  nonexistent
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• 22. Which of the following is the solution to the differential equation \( \dfrac{dy}{dx} = \dfrac{2y}{2x + 1} \) with the initial condition \( y(0) = e \) for \( x > -\dfrac{1}{2} \)?

  \( y = e^{2x^2 + 2x + 1} \)
  \( y = 2ex + e \)
  \( y = \sqrt{x^2 + x + e^2} \)
  \( y = \sqrt{\dfrac{1}{2}\ln(2x + 1) + e^2} \)
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• 23. The region enclosed by the graphs of \( y = x^2 \) and \( y = 2x \) is the base of a solid. For the solid, each cross section perpendicular to the \( y \)-axis is a rectangle whose height is 3 times its base in the \( xy \)-plane. Which of the following expressions gives the volume of the solid?

  ( 3 \displaystyle \int_{0}^{4} \left( \sqrt{y} - \frac{y}{2} \right)^2 dy \)
  \( 3 \displaystyle \int_{0}^{4} \left( \sqrt{y} + \frac{y}{2} \right)^2 dy \)
  \( 3 \displaystyle \int_{0}^{2} (2x - x^2)^2 dx \)
  \( 3 \displaystyle \int_{0}^{2} (2x + x^2)^2 dx \)
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• 24. If the average value of a continuous function \( f \) on the interval \([ -2, 4 ]\) is 12, what is \( \displaystyle \int_{-2}^{4} \frac{f(x)}{8} \, dx \) ?

  \( \frac{3}{2} \)
  \( 3 \)
  \( 9 \)
  \( 72 \)
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• 25. The figure above shows the graphs of the functions \( f \) and \( g \). If \( h(x) = f(x)g(x) \), then \( h'(2) = \)

  \( \frac{13}{2} \)
  \( \frac{1}{2} \)
  \( -1 \)
  \( -\frac{11}{2} \)
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• 26. \( \displaystyle \lim_{x \to \infty} \frac{\ln\left(e^{3x} + x\right)}{x} = \)

  \( 0 \)
  \( 1 \)
  \( 3 \)
  \( \infty \)
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• 27. The graph of \( f' \), the derivative of the function \( f \), is shown in the figure above. Which of the following statements must be true?

  I.  \( f \) is continuous on the open interval \( (a,b) \).

  II.  \( f \) is decreasing on the open interval \( (a,b) \).

  III.  The graph of \( f \) is concave down on the open interval \( (a,b) \).