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Integrals and Derivatives

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15.
A)
B)
C)
D) None of the above

16. What are the values of C0 and C1 in d(t) = C1 + C0t - 16t2, if d(1) = 4 and v(2) = -65?

A) C0 = -1, C1 = 21
B) C0 = 1, C1 = -21
C) C0 = -1, C1 = 19
D) C0 = 0, C1 = 1

17. What does du equal in ∫2x(x2 + 1)5 dx?

A) 2x
B) 2u du
C) 2x dx
D) 5u4

18. What is needed to fully determine an anti-differentiated function?
A) A lot of luck.
B) A boundary condition.
C) What its value is at (0, 0).
D) Its real world application.

19. What is the concavity of f(x) = 3x2 + ½x-1 + 2 @ (-1, 4½)?
A) 5
B) -5
C) 4½
D) 10½

20. What is the one thing done in anti-differentiation that has no counterpart in differentiation?

A) Adding a constant C.
B) Subtracting a constant C.
C) Dividing the new exponent by a constant C.
D) Nothing, they are equally matched step by step.

21.What is the primary difference between using anti-differentiation when finding a definite versus an indefinite integral?

A) Indefinite integrals don't have defined limits.
B) Definite integrals have defined limits.
C) The constant of integration, C.
D) There is no difference between definite and indefinite integrals.

22. What is the second step of performing anti-differentiation?

A) Divide the coefficient by the old exponential value.
B) Subtract the new exponential value from the coefficient.
C) Multiply the coefficient by the new exponential value.
D) Divide the coefficient by the new exponential value.

23. What is the value of R(-4) if:
, f (-4) = 2, h(-4) = 0, f '(-4) = 6, h'(-4) = 3.
A) 0
B) ½
C) 2
D) This is indeterminate.

24. ; what went wrong?

A) Nothing, integral calculus allows you to have negative areas.
B) Your math is wrong.
C) The function, 3x2 - 12 is below the x-axis along that interval, so it should have been negated first.
D) You had to differentiate the function first, and then fine the absolute value.

25. Which of the following is the best integration technique to use for ∫2x(x2 + 1)5 dx?

A) The product rule.
B) The chain rule.
C) The power rule.
D) The substitution rule.

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Control Systems - Proportional-Integral-Derivative Controlled Process and Nyquist Plots

Consider the block diagram attached, describing a process under Proportional-Integral-Derivative control.

1) Is the system open loop stable? Justify your answer

2) Let Ki = 10. Use the Routh-Hurwitz criterion to find the range of Kd, and Kp in terms of Kd, so that closed loop stability is achieved.

3) Suppose that Ki = 10, Kd = 1, and Kp = 15. A Bode plot of the open loop transfer function with these parameter values is shown in the attachment. Sketch a Nyquist plot for this system and use this sketch to determine if the system is closed loop stable.

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