Calculus

Question 1

What does δy/δx mean?

Answer 1

Small change

Question 2

Scalar multiple rule

Answer 2

d/dx[kf(x)]=kd/dx[f(x)]

Question 3

Addition/subtraction rule

Answer 3

d/dx[f(x)+-g(x)]=d/dx[f(x)]+-d/dx[g(x)]

Question 4

y=axⁿ

Answer 4

dy/dx=nax^n-1

Question 5

What is the equation of the tangent if the value of dy/dx at the point (x,y) is m?

Answer 5

y-y₁=m(x-x₁)

Question 6

What is the equation of the normal if the value of dy/dx at the point (x,y) is m?

Answer 6

y-y₁=-(1/m)(x-x₁)

Question 7

Variables x and y are connected by the equation y=3x²-4x. Find, in terms of p, the approximate change in y as x increases from 5 to 5+p where p is small

Answer 7

dy/dx=6x-4
dy/dx(x=5)=6(5)-4=26
δy/p=26
δy=26p

Question 8

y=e^x

Answer 8

dy/dx=e^x

Question 9

y=ln(x)

Answer 9

dy/dx=1/x

Question 10

y=tan(x)

Answer 10

dy/dx=sec²(x)

Question 11

Variables x and y are connected by y=2x³+2x. Given that x increases at a rate of 0.02 units per second, find the rate of change of y when x=3

Answer 11

0.02dy/dx=(6x²+2)*0.02
=0.12x²+0.04
0.02dy/dx(x=3)=0.12(3)²+0.04=1.12 units per second

Question 12

m(x)=f(g(x))

Answer 12

m^|=f^|(g(x))*g^|(x)

Question 13

m(x)=u(x)v(x)

Answer 13

m^|(x)=u^|(x)v(x)+u(x)v^|(x)

Question 14

m(x)=u(x)/v(x)

Answer 14

m^|(x)=(u^|(x)v(x)-u(x)v^|(x))/v²(x)

Question 15

At the maximum point

Answer 15

• dy/dx=0
• The gradient is positive to the left of the maximum and negative to the right
• d²y/dx²<0

Question 16

At the minimum point

Answer 16

• dy/dx=0
• The gradient is negative to the left of the minimum and positive to the right
• d²y/dx²>0

Question 17

dy/dx=xⁿ

Answer 17

y=xⁿ⁺¹/n+1 + c where c is a constant and n!=-1

Question 18

∫kf(x)dx

Answer 18

k∫f(x)dx where k is a constant

Question 19

∫[f(x)+-g(x)]dx

Answer 19

∫f(x)dx+-∫g(x)dx

Question 20

∫(ax+b)ⁿdx

Answer 20

(ax+b)ⁿ⁺¹/a(n+1) + c where n!=-1 and a!=0

Question 21

∫1/(ax+b)dx

Answer 21

(1/a)ln|ax+b| + c

Question 22

∫e^ax+bdx

Answer 22

(1/a)e^ax+b + c

Question 23

∫sec²(x)dx

Answer 23

tan(x) + c

Question 24

Find ∫²₁((x⁵+3)/x²)dx

Answer 24

=∫²₁x³+3x^-2 dx
= [x⁴/4+3x^-1/-1]²₁
= [x⁴/4-3/x]²₁
= 2⁴/4-3/2-(1⁴/4-3/1)=21/4

Question 25

What is area A, enclosed between the curves of the two functions f(x) and g(x), intersecting at x=a and x=b?

Answer 25

∫^b_af(x)dx-∫^b_ag(x)dx

Question 26

v

Answer 26

ds/dt

Question 27

a

Answer 27

dv/dt=d²s/dt²

Question 28

v

Answer 28

∫a dt

Question 29

s

Answer 29

∫v dt

Question 30

What can be found on a velocity-time graph?

Answer 30

- Acceleration is gradient - Displacement is area under

Question 31

What can be found on a displacement-time graph?

Answer 31

Velocity is gradient

Question 32

What can be found on an acceleration time graph?

Answer 32

Velocity is area under