C1: graphs, coordinates and differentiation

Question 1

How do you find the length of a line?

Answer 1

√((x1 - x2)^2 + (y1 - y2)^2))

Question 2

How do you find the gradient of a line?

Answer 2

(y2 - y1)/(x2 - x1)

Question 3

How do you find the midpoint of a line?

Answer 3

((x1 + x2)/2 , (y1 + y2)/2)

Question 4

How do you find the equation of a line?

Answer 4

y - y1 = m(x - x1)

Question 5

How do you find out if two lines are perpendicular?

Answer 5

m1m2 = -1

Question 6

Give the two forms of a circle equation.

Answer 6

(x - a)^2 + (y - b)^2 = R^2

x^2 + y^2 - 2ax - 2by + a^2 + b^2 - R^2 = 0

Question 7

y = f(x) -> y = f(x) + a

Answer 7

translation by vector (0 a)

moves up a units

Question 8

y = f(x) -> y = f(x + a)

Answer 8

translation by vector (-a 0)

moves left a units

Question 9

y = f(x) -> y = kf(x)

Answer 9

stretch parallel to y-axis with scale factor k

all y values multiplied by k

Question 10

y = f(x) -> y = f(kx)

Answer 10

stretch parallel to x-axis with scale factor 1/k

all x values divided by k

Question 11

y = f(x) -> y = -f(x)

Answer 11

reflection in x-axis

all y values negated

Question 12

y = f(x) -> y = f(-x)

Answer 12

reflection in y-axis

all x values negated

Question 13

y = x^n ; dy/dx = ?

Answer 13

dy/dx = nx^(n-1)

Question 14

f(x) = x^n ; f’(x) = ?

Answer 14

f’(x) = nx^(n-1)

nb DO NOT CONFUSE WITH f^-1(x)

Question 15

When dy/dx is positive, it can be said that y = x^n is…

Answer 15

…an increasing function.

Question 16

When does dy/dx = 0?

Answer 16

turning point (gradient = 0)

Question 17

Give the notation of the second derivative. (2)

Answer 17

d2y/dx2

f’‘(x)

Question 18

What information can the second derivative give?

Answer 18

> 0 stationary point is a minimum

< 0 stationary point is a maximum

Question 19

What do you do if the second derivative gives you 0?

Answer 19

test the gradient either side to judge; do not assume it is a point of inflection

Question 20

What is the difference between convex and concave?

Answer 20

convex: graph is below tangent lines
concave: graph is above tangent lines