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Feedback Control Systems Name: Homework 0 Time Limit:

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This exam contains 2 pages (including this cover page) and 3 questions. The total points are 30. The due date is 2/5(Fri), 6:00 pm. Submit it on blackboard 1. (10 points) Find the derivative of following functions. (a) (2 points) f (x) = 4x4 + 2x3 − 8x + 15 (b) (2 points) f (x) = 5x2 cos(x) (c) (2 points) f (x) = 5sin(x) − e5x (d) (2 points) f (x) = e3x sin(x2 ) (e) (2 points) f (x) =

x2 − 2 x3 + 3x − 2

2. (10 points) Find the integration of following functions. (a) (2 points)

Z

5

x3 − 4xdx

0

(b) (2 points)

(c) (2 points)

Z

3

2

ex dx −3

Z π 2 xsin(x)dx 0

(d) (2 points)

(e) (2 points)

[Hint for (e): Euler’s formula]

Z Z

3

loge xdx 1

0

ejx cos(x)dx −2

Feedback Control Systems

Homework 0 - Page 2 of 2

3. (10 points) Evaluate each of the integrals. δ(x) is delta function (a) (2 points)

Z

(b) (2 points)

Z

∞

3δ(t)dt 0

3

2

δ(t)2e4t dt

0

(c) (2 points)

Z

5

2

δ(t − 2)2e4t dt

3

(d) (2 points)

Z

∞ 2 (t3 )

δ(t)2e− tan

dt

0

(e) (2 points)

Z

0

3

2

δ(t − 2)2e4t dt...