Practise Test

1. \[ \int_{1}^{2} (4x^3 - x)\,dx = \]

2. Let \( f \) be the function defined by \[ f(x) = x^3 - 3x^2 - 9x + 11. \] At which of the following values of \( x \) does \( f \) attain a local minimum?

3. \[ \frac{d}{dx}\left(2(\sin\sqrt{x})^2\right) = \]

4. The position of a particle is given by the parametric equations \[ x(t) = \ln(t^2 + 1) \quad \text{and} \quad y(t) = e^{3 - t}. \] What is the velocity vector at time \( t = 1 \)?

5. \[ \sum_{n=1}^{\infty} \frac{e^n}{\pi^n} \text{ is} \]

6. The table above gives values of the twice-differentiable functions \( f \) and \( g \) and their derivatives at \( x = 2 \). If \( h \) is the function defined by \( h(x) = \dfrac{f'(x)}{g(x)} \), what is the value of \( h'(2) \)?

7. Which of the following is the Maclaurin series for \( x\cos(x^2) \)?

8. \[ \lim_{x \to \infty} \frac{10 - 6x^2}{5 + 3e^x} \text{ is} \]

9. The function \( f \) is not differentiable at \( x = 5 \). Which of the following statements must be true?

10. The second derivative of a function \( f \) is given by \[ f''(x) = x(x - 3)^5(x - 10)^2. \] At which of the following values of \( x \) does the graph of \( f \) have a point of inflection?

11. \[ \int_{0}^{\pi} \frac{e^x - 1}{e^x - x}\,dx = \]

12. The table above gives selected values for the differentiable function \( f \). In which of the following intervals must there be a number \( c \) such that \( f'(c) = 2 \)?

13. Let \( y = f(x) \) be the solution to the differential equation \[ \frac{dy}{dx} = x + 2y \] with initial condition \( f(0) = 2 \). What is the approximation for \( f(-0.4) \) obtained by using Euler’s method with two steps of equal length starting at \( x = 0 \)?

14. What is the slope of the line tangent to the curve \( \sqrt{x} + \sqrt{y} = 2 \) at the point \( \left(\dfrac{9}{4}, \dfrac{1}{4}\right) \)?

15. \[ \int_{1}^{\infty} \frac{6}{(x + 3)^{3/2}}\,dx \text{ is} \]

16. If \[ \frac{dy}{dx} = 2 - y, \] and if \( y = 1 \) when \( x = 1 \), then \( y = \)

17. Which of the following series converges?

18. Let \( R \) be the region in the first and second quadrants between the graphs of the polar curves \( f(\theta) = 3 + 3\cos\theta \) and \( g(\theta) = 4 + 2\cos\theta \), as shaded in the figure above. Which of the following integral expressions gives the area of \( R \)?

19. Which of the following statements about the series \[ \sum_{n=1}^{\infty} \frac{(-1)^n n}{n^2 + 1} \] is true?

20. If \[ \int_{1}^{4} f(x)\,dx = 8 \quad \text{and} \quad \int_{1}^{4} g(x)\,dx = -2, \] which of the following cannot be determined from the information given?

21. Which of the following is the interval of convergence for the series \[ \sum_{n=1}^{\infty} \frac{(x + 4)^n}{n \cdot 5^{n+1}} \; ? \]

22. What is the slope of the line tangent to the polar curve \[ r = 3\theta \] at the point where \[ \theta = \frac{\pi}{2} \; ? \]

23. The definite integral \[ \int_{0}^{4} \sqrt{x}\,dx \] is approximated by a left Riemann sum, a right Riemann sum, and a trapezoidal sum, each with 4 subintervals of equal width. If \( L \) is the value of the left Riemann sum, \( R \) is the value of the right Riemann sum, and \( T \) is the value of the trapezoidal sum, which of the following inequalities is true?

24. The graph of the piecewise linear function \( f \) is shown above. What is the value of \[ \int_{0}^{12} f'(x)\,dx \; ? \]

25. The function f has a continuous derivative. If \( f(0) = 1 \), \( f(2) = 5 \), and \( \int_{0}^{2} f(x)\,dx = 7 \), what is \( \int_{0}^{2} x \cdot f'(x)\,dx \)?

26. Which of the following series are conditionally convergent? I.  \( \displaystyle \sum_{n=1}^{\infty} \frac{(-1)^n}{n!} \) II.  \( \displaystyle \sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}} \) III.  \( \displaystyle \sum_{n=1}^{\infty} \frac{(-1)^n}{n+2} \)

27. Let R be the region in the first quadrant bounded by the graph of \( y = \sqrt{x - 1} \), the x-axis, and the vertical line \( x = 10 \). Which of the following integrals gives the volume of the solid generated by revolving R about the y-axis?

28. If \( \dfrac{dx}{dt} = 5 \) and \( \dfrac{dy}{dt} = \sin(t^2) \), then \( \dfrac{d^2y}{dx^2} \) is

29. Which of the following expressions is equal to \( \displaystyle \lim_{n \to \infty} \frac{1}{n} \left( \frac{1}{2 + \frac{1}{n}} + \frac{1}{2 + \frac{2}{n}} + \frac{1}{2 + \frac{3}{n}} + \cdots + \frac{1}{2 + \frac{n}{n}} \right) \)?

30. A function f has a Maclaurin series given by \( 2 + 3x + x^2 + \frac{1}{3}x^3 + \cdots \), and the Maclaurin series converges to \( f(x) \) for all real numbers \( x \). If g is the function defined by \( g(x) = e^{f(x)} \), what is the coefficient of \( x^2 \) in the Maclaurin series for g?

1. The function f is continuous on the closed interval \( [0, 5] \). The graph of \( f' \), the derivative of f, is shown above. On which of the following intervals is f increasing?

2. If \( \dfrac{dy}{dt} = 6e^{-0.08(t - 5)^2} \), by how much does \( y \) change as \( t \) changes from \( t = 1 \) to \( t = 6 \)?

3. The function f has the property that \( \lim_{x \to 1^-} f(x) = +\infty \), \( \lim_{x \to 1^+} f(x) = -\infty \), \( \lim_{x \to -\infty} f(x) = 2 \), and \( \lim_{x \to +\infty} f(x) = 2 \). Of the following, which could be the graph of f?

4. Tara’s heart rate during a workout is modeled by the differentiable function h, where \( h(t) \) is measured in beats per minute and \( t \) is measured in minutes from the start of the workout. Which of the following expressions gives Tara’s average heart rate from \( t = 30 \) to \( t = 60 \)?

5. The function f has derivatives of all orders for all real numbers with \( f(2) = -1 \), \( f'(2) = 4 \), \( f''(2) = 6 \), and \( f'''(2) = 12 \). Using the third-degree Taylor polynomial for f about \( x = 2 \), what is the approximation of \( f(2.1) \)?

6. Let \( f \) be a function with derivative given by \( f'(x) = \sqrt{x^3 + 1} \). What is the length of the graph of \( y = f(x) \) from \( x = 0 \) to \( x = 1.5 \)?

7. Let \( h \) be a continuous function of \( x \). Which of the following could be a slope field for a differential equation of the form \( \dfrac{dy}{dx} = h(x) \)?

8. \[ f(x) = \begin{cases} k^3 + x & \text{for } x < 3 \\ \dfrac{16}{k^2 - x} & \text{for } x \ge 3 \end{cases} \] Let \( f \) be the function defined above, where \( k \) is a positive constant. For what value of \( k \), if any, is \( f \) continuous?

9. Let \( f \) be a differentiable function. The figure above shows the graph of the line tangent to the graph of \( f \) at \( x = 0 \). Of the following, which must be true?

10. A referee moves along a straight path on the side of an athletic field. The velocity of the referee is given by \( v(t) = 4(t - 6)\cos(2t + 5) \), where \( t \) is measured in minutes and \( v(t) \) is measured in meters per minute. What is the total distance traveled by the referee, in meters, from time \( t = 2 \) to time \( t = 6 \)?

11. The graph of the function \( f \) shown above consists of three line segments. Let \( h \) be the function defined by \( h(x) = \int_{0}^{x} f(t)\,dt \). At what value of \( x \) does \( h \) attain its absolute maximum on the interval \( [-4, 3] \)?

12. The position of a particle moving in the \( xy \)-plane is given by the parametric equations \( x(t) = e^{-t} \) and \( y(t) = \sin(4t) \) for time \( t \ge 0 \). What is the speed of the particle at time \( t = 1.2 \)?

13. Let \( f \) be the function defined by \[ f(x) = \frac{1}{4}x^4 - \frac{2}{3}x^3 + \frac{1}{2}x^2 - \frac{1}{2}x. \] For how many values of \( x \) in the open interval \( (0, 1.565) \) is the instantaneous rate of change of \( f \) equal to the average rate of change of \( f \) on the closed interval \( [0, 1.565] \)?

14. The population \( P \) of rabbits on a small island grows at a rate that is jointly proportional to the size of the rabbit population and the difference between the rabbit population and the carrying capacity of the population. If the carrying capacity of the population is 2400 rabbits, which of the following differential equations best models the growth rate of the rabbit population with respect to time \( t \), where \( k \) is a constant?

15. A region is bounded by two concentric circles, as shown by the shaded region in the figure above. The radius of the outer circle, \( R \), is increasing at a constant rate of 2 inches per second. The radius of the inner circle, \( r \), is decreasing at a constant rate of 1 inch per second. What is the rate of change, in square inches per second, of the area of the region at the instant when \( R \) is 4 inches and \( r \) is 3 inches?