Differentiation

Question 1

Limit

Answer 1

the value a function approaches, when its variable tends towards a certain value

Question 2

df/dx = f’(x) =

Answer 2

lim f(x+h) - f(x) / h
h–>0

Question 3

Using first principles, find the derivative of f(x) = x^3 - 4x.

Answer 3

f(x+h) = (x+h)^3 - 4(x+h)
= x^3 + 3hx^2 + (3h^2 - 4)x + h^3 - 4h

lim h–> (3x^2 + 3hx + h^2 - 4)
= 3x^2 - 4

Question 4

d/dx (ax^n) =

Answer 4

anx ^ n-1

Question 5

For sums of different terms,
differentiate

Answer 5

each term seperately

Question 6

.

Answer 6

.

Question 7

For products or quotients,
do not differentiate

Answer 7

simplify first

Question 8

y = x^4(x^5 - 3x^-2) + 2 differentiate

Answer 8

9x^8 - 6x

Question 9

To find equations of tangents/normals at a point

Answer 9

use differentiation to find and evaluate the gradient of the curve, then substitute into y - y1 = m(x - x1)

Question 10

A curve has equation y = x^3 - 5x^2 - x^3/2 + 22
tangent and normal at the point (4, -2)

Answer 10

tangent
y = 5x-22
normal
y = -x/5 - 6/5

Question 11

A curve has equation y = a/x.
The normal to the curve at x = 3 is parallel to x - 3y + 6 = 0.
Find the value of the constant a.

Answer 11

m = 1/3
gradient is -3
a = 27

Question 12

rate of change at f’(x) > 0

Answer 12

increasing

Question 13

rate of change at f’(x) < 0

Answer 13

decreasing

Question 14

rate of change at f’(x) = 0

Answer 14

stationary

Question 15

rate of change of the gradient

Answer 15

second derivative
d2y/dx2

Question 16

Identify where f(x) = ¼x^4 + ⅓x^3 - 3x^2 is increasing

Answer 16

x(x+3)(x-2)
increasing –> x > 2 , -3<x<0

Question 17

d2y / dx2 < 0

Answer 17

local maxima

Question 18

d2y / dx2 > 0

Answer 18

local minima

Question 19

d2y / dx2 = 0

Answer 19

no conclusion

Question 20

A point of inflection is

Answer 20

where the 2nd derivative changes sign - always d2y/d2x = 0

Question 21

to find stationary points

Answer 21

find derivative
solve derivative = 0
use x to work out y
find second derivative
evaluate at each coord

Question 22

For the curve y = x^3 - 12x + 3, find and classify
the coordinates of the stationary points.

Answer 22

d2y/dx2 = 6x
(2,-13) is a minimum
(-2,19) is a maximum

Question 23

Optimisation

Answer 23

form equations
differentiate twice
find stationary points
classify them
answer question

Question 24

A wire of length 12 cm is bent to form a rectangle.
Show that the area, A, is given by A = 6x - x2, where x is the width of the rectangle
Find the maximum possible area

Answer 24

area + 6x - x^2
max area when d2A/dx2 < 0
0 = 6-2x , x=3
d2A/dx2 = -2 < 0 so max at x+3
max area +6x3-9 = 9cm

Question 25

An open cuboid measures x by 2x by h units.
Its inner surface area is 12 units.
Show that the volume V is
given by V = ⅔x(6 - x^2)

Find the exact value of x for
which V is maximised

Answer 25

SA = 2xh + 4xh + 2x^2
12 = 6xh + 2x^2
h + 12-2x^2 / 6x
vol = 2x^2h = 2/3x (6-x^2)

max V –> d2V/dx2 < 0
dV/dx = 0 = 4-2x^2
x = +- root 2
d2V/dx^2 = -4x
will be negative for x = root 2