MTH U343 QUIZ 12 Answers

  1. Calculate the following.

    1. MATH

      The basic function here is $t^{2}$ whose transform is $\dfrac{2}{s^{3}}$; however, the $e^{-3t}$ multiplier creates a translation (see table), resulting in the answer MATH.

    2. MATH

      Since the polynomial doesn't factor, we complete the square, giving MATH. The basic function here is $\sin 5t$, but the $s+4$ means there was a translation; we also need a $5$ in the numerator: MATH

    3. MATH

      Forget the $e^{-2s}$ till the end. The inverse-transform of $\dfrac{1}{s^{2}} $ is $t$ our "basic" function here. Multiplying now by the $e^{-2s}$ causes a translation (see formulas), resulting in the answer $u(t-2)\cdot (t-2)$.

    4. MATH

      Once again, leaving the $e^{-7s}$ for last, we see that the basic function here is is $\dfrac{1}{s+5}$ whose inverse transform is $e^{-5t}$. Returning to the $e^{-7s}$ we see that we need a translation of the basic function by $7$, resulting in the answer $u(t-7)e^{-5(t-7)}$.

  2. Let MATH.

    1. Explain why MATH.

      When $0\leq t<\pi $, $u(t-\pi )=0$, so MATH.

      When $t\geq \pi $, $u(t-\pi )=1$, so MATH. Thus, the function agrees with $f(t)$.

    2. Using part a, calculate MATHMATH

    3. Let MATH. Calculate MATH

    The easiest way to do this is to note that MATH soMATHAnother approach is to notice directly that MATH, and also MATH.







Laplace Transforms




MATH

MATH




MATH