Free Printable Properties of Laplace Transform

1.

Laplace\ transform\ of\ e^{3t}-\frac{1}{2}

a)

$\frac{1}{s+3}-\frac{1}{2s}$

b)

$\frac{1}{s-3}-\frac{2}{s}$

c)

$\frac{1}{s-3}-\frac{1}{2s}$

d)

$\frac{1}{2\left(s-3\right)}$

2.

L\left[e^{-2t}\sin3t\right]

a)

$L\left[\sin3t\right],\ s\rightarrow s+2$

b)

$L\left[\sin3t\right],\ s\rightarrow s-2$

3.

f\left(t\right)=te^{-2t}

a)

b)

c)

d)

4.

L\left[e^{5t}\cos3t\right]

a)

b)

c)

d)

5.

Define this property: $L\left(f\left(t\right)e^{at}\right)=F\left(s-a\right)$

a)

Linearity Property

b)

First Shifting Property

c)

Convolution Theorem

d)

Second Shifting Property

6.

$L\left(e^{2t\ }\sin\ t\right)$ What is $a$ and f(t)?

a)

$a=2,\ f\left(t\right)=e^{2t}$

b)

$a=4,\ f\left(t\right)=\cos\ t$

c)

$a=2,\ f\left(t\right)=\sin\ t$

7.

Find $L\left(e^{3t}\left(t^3-3t^2+5t\right)\right)$

a)

$\frac{2}{\left(s-3\right)^4}-\left(\frac{3}{\left(\left(s-3\right)^3\right)}\right)+\frac{5}{\left(s-3\right)^2}$

b)

$\frac{6}{\left(s-3\right)^4}-\left(\frac{6}{\left(s-3\right)^3}\right)+\frac{5}{\left(s-3\right)^2}$

c)

$\frac{2}{\left(s-3\right)^3}-\left(\frac{3}{\left(s-3\right)^{\left(2\right)}}\right)+\frac{5}{s-3}$

8.

$L\left(t\ \cos\ 6t\right)$ Which Laplace property is use to solve this?

a)

Linearity Property

b)

Derivative of Laplace Transform

c)

First Shifting Property

d)

Second Shifting Property

9.

$L\left(t\ \sin\ 4t\right)$ What is n, f(t) and L[f(t)]?

a)

$n=1,\ f\left(t\right)=\sin\ 4t,\ F\left(s\right)=\frac{4}{s^2+16}$

b)

$n=4,\ f\left(t\right)=t,\ F\left(s\right)=\frac{4}{s^2+16}$

c)

$n=1,\ f\left(t\right)=\cos\ 6t,\ F\left(s\right)=\frac{s}{s^2+14}$

10.

What is the formula for Laplace First Order Derivative?

a)

$ℒ\left(𝑓′(𝑡)\right)=𝑠𝐹(𝑠)−𝑓(0)$

b)

$ℒ\left(𝑓"(𝑡)\right)=𝑠^2𝐹(𝑠)−𝑠𝑓\left(0\right)−𝑓′\left(0\right)$

c)

$ℒ\left(𝑓′(𝑡)\right)=𝑠^2𝐹(𝑠)−𝑓(0)$

11.

What is the formula for Laplace Second Order Derivative?

a)

$ℒ\left(𝑓′(𝑡)\right)=𝑠𝐹(𝑠)−𝑓(0)$

b)

$ℒ\left(𝑓"(𝑡)\right)=𝑠^2𝐹(𝑠)−𝑠𝑓\left(0\right)−𝑓′\left(0\right)$

c)

$ℒ\left(𝑓"(𝑡)\right)=𝑠^2𝐹(𝑠)−𝑓(0)$

12.

Given $g\left(t\right)=e^{5t}f\left(t\right)$ , what is G(s)?

a)

$G\left(s\right)=F\left(s-5\right)$

b)

$G\left(s\right)=\left(-1\right)^5\frac{\text{d}^5F\left(s\right)}{\text{d}s^5}$

c)

$G\left(s\right)=F\left(s+5\right)$

d)

$G\left(s\right)=\left(-1\right)^4\frac{\text{d}^4F\left(s\right)}{\text{d}s^4}$

13.

Given $g\left(t\right)=t^3f\left(t\right)$ , what is G(s)?

a)

$G\left(s\right)=F\left(s-3\right)$

b)

$G\left(s\right)=\left(-1\right)^3\frac{\text{d}^3F\left(s\right)}{\text{d}s^3}$

c)

$G\left(s\right)=F\left(s+3\right)$

d)

$G\left(s\right)=\left(-1\right)^2\frac{\text{d}^2F\left(s\right)}{\text{d}s^2}$

14.

f(t)=sinh 4t, F(s)=?

a)

$\frac{a}{s^2-a^2}$

b)

$\frac{a}{s^2+a^2}$

c)

$\frac{4}{s^2-16}$

d)

$\frac{4}{s^2+16}$

15.

Given $g\left(t\right)=\frac{f\left(t\right)}{t}$ , what is G(s)?

a)

$G\left(s\right)=\int_s^{\infty}F\left(u\right)du$

b)

$G\left(s\right)=\left(-1\right)\frac{\text{d}F\left(s\right)}{\text{d}s}$

c)

$G\left(s\right)=\int_{\infty}^sF\left(u\right)du$

d)

$G\left(s\right)=^{\left(-1\right)^{-1}}\frac{\text{d}F\left(s\right)}{\text{d}s}$

16.

Given $g\left(t\right)=e^{7t}\ \ \cosh2t$ . Which function is selected to be transformed into s-domain first?

a)

$\cosh2t$

b)

$e^{7t}$

17.

Given $g\left(t\right)=t^2\ \ \cos3t$ . Which function is selected to be transformed into s-domain first?

a)

$\cos3t$

b)

$t^2$

Properties of Laplace Transform

Total questions: 17

Worksheet time: 17mins

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