Free Printable 3.1 Laplace Transform

1.

Laplace Transform transforms a function from

a)

$f\left(t\right)\ to\ F\left(s\right)$

b)

$F\left(s\right)\ to\ f\left(t\right)$

c)

$f\left(t\right)\ to\ f'\left(t\right)$

d)

$f"\left(t\right)\ to\ f'\left(t\right)$

2.

Which one is the correct definition of Laplace Transform?

a)

$F\left(s\right)=\int_0^{\infty}e^{-st}f\left(t\right)dt$

b)

$F\left(s\right)=\int_0^tf\left(t\right)g\left(t-u\right)$

3.

What is the Laplace Transform for $f\left(t\right)=t^3$ ?

a)

$\frac{1}{s^3}$

b)

$\frac{2}{s^3}$

c)

$\frac{6}{s^4}$

d)

$\frac{3}{s^3}$

4.

What is Laplace Transform for $e^{-2t}$ ?

a)

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

b)

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

c)

$\frac{2}{s}$

d)

$\frac{1}{s^2}$

5.

What is Laplace Transform for $\sin\ 2t$ ?

a)

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

b)

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

c)

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

d)

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

6.

What is Laplace Transform for $6\ \sin\ 2t$ ?

a)

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

b)

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

c)

$\frac{6}{s^2+4}$

d)

$\frac{12}{s^2+4}$

7.

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

8.

$L\left(e^{3t}t\right)$ Based on first shifting property, what is $a$ value?

a)

1

b)

2

c)

3

d)

4

9.

Find $L\left(e^{3t}t\right)$

a)

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

b)

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

c)

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

d)

$\frac{1}{\left(s\right)}$

10.

$L\left(e^{2t\ }\sin\ t\right)$ 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)

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

11.

Find $L\left(\cos\ t\right)$

a)

$\frac{1}{s^2+1^{ }}$

b)

$\frac{1}{s^2+4}$

c)

$\frac{s}{s^2+1}$

12.

Find $L\left(e^{2t}\sin\ 2t\right)$

a)

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

b)

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

c)

$\frac{s}{\left(s-4\right)^2+1}$

13.

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

a)

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

b)

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

c)

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

14.

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)

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

15.

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

a)

Linearity Property

b)

Derivative of Laplace Transform

c)

First Shifting Property

d)

Second Shifting Property

16.

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

a)

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

b)

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

c)

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

17.

$Find\ L\left(t^{ }\sin\ 4t\right)$

a)

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

b)

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

c)

$-\frac{8s}{\left(s^2+16\right)^2}$

18.

$L\left(\sin^2t\right)$ What is $f\left(t\right),\ f'\left(t\right)\ and\ f\left(0\right)$ ?

a)

$f\left(t\right)=\sin^2t,\ f'\left(t\right)=2\ \cos t,\ f\left(0\right)=0$

b)

$f\left(t\right)=\sin^2t,\ f'\left(t\right)=\sin\ 2t,\ f\left(0\right)=0$

c)

$f\left(t\right)=\cos^2t,\ f'\left(t\right)=-\ \sin2t,\ f\left(0\right)=1$

19.

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)

$ℒ\left(𝑓′(𝑡)\right)=𝑠^2𝐹(𝑠)−𝑓(0)$

20.

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)

$ℒ\left(𝑓"(𝑡)\right)=𝑠^2𝐹(𝑠)−𝑓(0)$

3.1 Laplace Transform

Total questions: 20

Worksheet time: 10mins

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