Winter Exam

Question 1

f(x) –> f(x+a)+b

Answer 1

Translation of (-a,b)

Question 2

f(x) –> bf(ax)

Answer 2

Stretch 1/a parallel to x, stretch b parallel to y

Question 3

In f(ax+b), which operation is done first?

Answer 3

The translation!

Question 4

f(x) –> -f(x)

Answer 4

Reflection in x axis

Question 5

f(x) –> f(-x)

Answer 5

Reflection in y axis

Question 6

f^-1(x)

Answer 6

Inverse function, maps input onto output (basically reflection in y=x)

Question 7

What makes a function even? (3)

Answer 7

Symmetry at y axis, all even powers with constants allowed, f(x)=f(-x)

Question 8

What makes a function odd?

Answer 8

Rotational symmetry order 2 about origin, all odd powers and no constants, f(-x)=-f(x)

Question 9

You know what?

Answer 9

A worthy… ;)

Question 10

What would you use to differentiate a function of a function?

Answer 10

Chain rule, dy/dy = dy/du x du/dx.
The easy one basically

Question 11

What would you use to differentiate two multiplied functions? (y=uv)

Answer 11

Product rule, dy/dx = v(du/dx) + u(dv/dx)

Question 12

What would you use to differentiate two divided functions? (y=u/v)

Answer 12

Quotient rule, dy/dx = (v(du/dx) - u(dv/dx))/v^2

Question 13

How do you differentiate when x is the subject?

Answer 13

dy/dx = 1/dx/dy

Question 14

d/dx e^ax

Answer 14

ae^ax

Question 15

d/dx lnax

Answer 15

1/x

Question 16

d/dx ln(f(x))

Answer 16

f’(x)/f(x)

Question 17

d/dx lnx^a

Answer 17

a/x

Question 18

d/dx ae^x

Answer 18

ae^x

Question 19

d/dx e^f(x)

Answer 19

f’(x)e^f(x)

Question 20

d/dx sinkx

Answer 20

kcoskx

Question 21

d/dx coskx

Answer 21

-ksinkx

Question 22

1 + tan^2 x

Answer 22

-sec^2(x)

Question 23

d/dx a^kx

Answer 23

ka^kx lna

Question 24

Integration by substitution: 4 steps

Answer 24

1. define u
2. work out limits in terms of u
3. work out du/dx and rearrange as needed
4. proceed with integration!

Question 25

Integral of e^ax

Answer 25

1/a e^ax + c

Question 26

Integral of 1/x

Answer 26

lnx + c

Question 27

Integral of f’(x)/f(x)

Answer 27

ln|f(x)| + c

Question 28

Integral of sinax

Answer 28

-1/a cosax + c

Question 29

Integral of cosax

Answer 29

1/a sinax + c

Question 30

Integral of 1/ax+b

Answer 30

1/a ln|ax+b| + c

Question 31

Integral of U(du/dx) =

Answer 31

UV - (int)V(du/dx) dx

Question 32

sec(x)

Answer 32

1/cos(x)

Question 33

cosec(x)

Answer 33

1/sin(x)

Question 34

cot(x)

Answer 34

1/tan(x)

Question 35

1+cot^2 (x)

Answer 35

cosec^2 (x)

Question 36

1+tan^2 (x)

Answer 36

sec^2 (x)

Question 37

sin(A+-B)

Answer 37

sinAcosB+-cosAsinB

Question 38

cos(A+-B)

Answer 38

cosAcosB-+sinAsinB

Question 39

tan(A+-B)

Answer 39

(tanA+-tanB)/(1-+tanAtanB)

Question 40

cos2A

Answer 40

2cos^2 A - 1 and 1-2sin^2 A

Question 41

For small positive x

Answer 41

sin(x)~~tan(x)~~(x)
cos(x)~~(1-(x)^2)/2

Question 42

Guess what

Answer 42

You’re amazing! :)