Pure

Question 1

steps to find nature of stationary points

Answer 1

• find the derivative
• make it equal to zero
• solve to get stationary points
• find derivative of derivative
• sub in stationary points into second derivative
• positive = minimum point, negative = maximum point

Question 2

vertical asymptote is found by

Answer 2

makind the denominator equal to 0

Question 3

horizontal asymptote is found by

Answer 3

making the x value really large

Question 4

When finding antiderivative, remember:

Answer 4

to add c (where c is any constant)

Question 5

how to sketch a v-t graph given a d-t graph

Answer 5

velocity is the derivative of displacement

Question 6

how to use calculus to find the turning point of a quadratic:

Answer 6

• find derivative of quadratic
• solve for x
• sub x into quadratic equation to find y

Question 7

equation for sum to Infinity In a convergent series.

Answer 7

S∞ = a / (1 - r)

Question 8

Fully describe the transformation that maps the curve with equation y = x² - 3 onto the curve with equation y = 2x² - 6
[3 marks]

Answer 8

Stretch;
Parallel to y-axis;
Scale factor 2;

Question 9

b3 points through a circle

PQ is the diameter of the circle, since

Answer 9

angle subtended at circumference is 90°

Question 10

The equation of a curve is y=4x²+8x+1.
The curve is stretched parallel to the x-axis with scale factor 2.
Find the equation of the new curve, giving your answer in the form y=ax²+bx+c , where a, b and c are integers to be determined. [2]

→

Answer 10

• to stretch parallel to x axis: divide each x term by 2
• y = 4 (x/2)² + 8 (x/2) + 1
• = x² + 4x + 1

Question 11

you CANNOT

Answer 11

log a negative number;

Question 12

derivative of csc x

Answer 12

-cosec(x)cot(x)

Question 13

derivative of secx

Answer 13

sec(x)tan(x)

Question 14

derivative of cotx

Answer 14

-csc²x

Question 15

derivative of cos(x)

Answer 15

-sin(x)

Question 16

derivative of tanx

Answer 16

sec²x

Question 17

derivative of sec²x

Answer 17

2sec²xtanx

Question 18

Derived identities using sin²x + cos²x = 1 :

Answer 18

dividing by cos²x
tan²x + 1 = sec²x

dividing by sin²x
1 + cot²x = cosec²x

Question 19

How small does angle have to be in small angle approximations?

Answer 19

• angle should be measured in radians
• In small angle approximations, the angle (in radians) should be small enough that sin(θ) ≈ θ and cos(θ) ≈ 1

Typically, this is valid for angles less than about 0.1 radians (approximately 5.7 degrees)

Question 20

radian to degrees:

Answer 20

π radians = 180°
1 radian = 180° / π

Question 21

horizontal asymptotes:

Answer 21

• degree on top < degree on bottom then y = 0
• both same: take the highest powers and divide them eg. 2x^2 / 3x^2 = 2/3