9 differentiation

Question 1

How do you do a show that question that makes you express an area or a volume?

Answer 1

• Create an equation for what you know

- Create an equation which lets you eliminate what is not in the equation.

Question 2

how do you differentiate a function that contains y?

Answer 2

• D/dx with arrows going down.
• Treat the x’s normally
• Anything that is attached to a y must have dy/dx next to it.
• Make dy/dx the subject

Question 3

How do you justify why your answer is a min or max?

Answer 3

• Min point second derivative > 0

- Max point second derivative <0

Question 4

What is the difference between the tangent and the normal to point a?

Answer 4

-Tangent has the same gradient of the curve it meets, normal is perpendicular to curve it meets.

Question 5

What is an increasing and decreasing function?

Answer 5

• Increasing function f’(x)>0

- Decreasing function f’(x)<0

Question 6

What can a second derivative tell you?

Answer 6

f’‘(x)>0 local minimum
f’‘(x)< local maximum
f’‘(x)= 0 max, min, or point of inflexion (requires table to be drawn out).

Question 7

How can you investigate f’‘(x)=0

Answer 7

• Look at the gradients either side

- If the gradient is the same sign, point of inflexion.

Question 8

What does dy/dx represent?

What else can you think of it as?

Answer 8

• The rate of change of y with respect to x

- Small change in y over small change in x.

Question 9

what are the features of f(x) to f’(x) that can be used to sketch.

Answer 9

f(x) max or min ——–> f’(x) cuts the x axis
f(x) point of inflexion —-> f’(x) touches the x axis.
positive gradient —–> above x axis
negative gradient ——> below x axis.
Vertical asymptote = vertical asymptote.

Question 10

What must be remembered when calculating the maximum value of something in a modelling question?

Answer 10

-sub in the max value that it took when dy/dx equaled 0.

Question 11

How do you prove that a point is a point of inflexion?

Answer 11

-Show that the gradient either side dy/dx<0

Question 12

How would you differentiate something in the form (x+3)^2?

Answer 12

Multiply by the power, multiply by the derivative of the bracket and reduce the power by 1.

Question 13

How would you differentiate something in the form xsinx?

Answer 13

Product rule.
make u=x make v=sinx
du/dx=1 dv/dx = cosx
Then multiply u by dv/dx and vice versa

Question 14

How can you differentiate trig functions that are not sin or cos?

Answer 14

-Use formula booklet page 6

Question 15

How do you prove the derivative of cosec?

Answer 15

• Rewrite as (sin)^-1
• Then use chain rule
• Then manipulate answer.

Question 16

How can you find the gradient of a parametric function?

Answer 16

if given in x=t and y=t
work out dx/dt and dy/dt.
Then multiply together to get dy/dx

Question 17

How can you work out the turning point of implicit differential function?

Answer 17

-Make it equal zero then substitute into

Question 18

How do you know if a function is concave or convex?

Answer 18

• ConCAVE if f’‘(x)<0

- Convex if f’‘(x)>0

Question 19

What is another way to prove a point of inflexion?

Answer 19

f’‘(x)=0 at that point and has opposing signs either side.

Question 20

What information is useful to pick out in rates of change questions?

How can you go about solving them?

Answer 20

-if units are shows as m^3min that shows it is dv/dt

• Extract as much information possible to form two equations.
• Then combine these two equations