Differentiation (P1.12 & P2.9)

Question 1

what is the gradient of a curve at a given point defined as?

Answer 1

the gradient of the tangent to the curve at that point

Question 2

as point B moves closer to point A on the curve, what happens to the gradient of chord AB?

Answer 2

gets closer to the gradient of the tangent to the curve at A

Question 3

label graph with x0, x0+h & the coordinates of A & B

Answer 3

pg 259 P1 textbook

Question 4

differentiate from first principles

Question 5

what is the derivative of the quadratic ax^2 + bx + c?

Answer 5

straight line with the gradient 2a
crosses the x-axis once at the point where the quadratic curve has 0 gradient = the turning point of the quadratic graph

Question 6

what is the equation of the normal to the curve y=f(x) at the point with coordinates (a, f(a))?

Answer 6

y-f(a) = -1/f’(a) (x-a)
like y-y1 = m(x-x1)

Question 7

describe how to use the derivative to determine whether a function is increasing or decreasing on a given interval

Answer 7

the function f(x) is increasing on the interval [a,b] if f’(x)≥0 when a<x<b

the function is decreasing when f’(x)≤0

Question 8

what does differentiating a function twice show?

Answer 8

the rate of change of the gradient
= second order derivative

Question 9

f’‘(x) =

Answer 9

d^2y / dx^2

Question 10

what is a stationary point & different types?

Answer 10

any point where the curve has gradient zero
local maximum, local minimum, point of inflection

Question 11

complete table of type of stationary point, f’(x-h), f’(x) & f’(x+h)

Answer 11

see P1 textbook pg 273

Question 12

show the different types of stationary points graphically

Answer 12

see P1 pg 273

Question 13

how can the second derivative be used to determine the nature of a stationary point?

Answer 13

if function f(x) has stationary point at x=a
f’‘(a) > 0 point is a local minimum
f’‘(a) < 0 point is a local maximum
f’‘(a) = 0 point could be local max, min or point of inflection –> must look at points either side to determine its nature

Question 14

how can second derivatives be used to determine whether a curve is concave or convex on a given domain?

Answer 14

concave: f’‘(x) ≤ 0 looks like a cave on graph ∩
convex: f’‘(x) ≥ 0 looks like the V on graph ∪

Question 15

define point of inflection

Answer 15

the point at which a curve changes from being concave to convex or vice versa
= the point at which f’‘(x) changes sign

Question 16

when sketching gradient functions, what features of the graph y=f’(x) correspond with the following y=f(x) features?
maximum/minimum
point of inflection
+ve gradient
-ve gradient
vertical asymptote
horizontal asymptote

Answer 16

cuts the x-axis
touches the x-axis
above the x-axis
below the x-axis
vertical asymptote
horizontal asymptote at x-axis

Question 17

differentiate y = sinx from first principles

Question 18

differentiate y = cosx from first principles

Question 19

what small angle approximations are used in differentiating sinx & cosx from first principles?

Answer 19

sinx ~ x
cosx ~ 1 - 1/2(x^2)

Question 20

what do sinx & cosx differentiate to?

Answer 20

sinx –> cosx
cosx –> -sinx

Question 21

f(x) = ln(f(x))
f’(x) =

Answer 21

f’(x)/f(x)

Question 22

f(x) = a^kx
f’(x) =

Answer 22

a^kx . k . lna

Question 23

chain rule

Answer 23

dy/dx = dy/du x du/dx
f’(g(x))g’(x)
n(f(x))^n-1f’(x)

Question 24

product rule f(x)g(x)

Answer 24

f’(x)g(x) + f(x)g’(x)

Question 25

quotient rule u/v

Answer 25

vu’ - uv’ / v^2

Question 26

parametric differentiation equation

Answer 26

dy/dx = dy/dt / dx/dt

Question 27

d/dx (f(y)) =

Answer 27

f’(y) dy/dx

Question 28

d/dx (y^n) =

Answer 28

ny^n-1 dy/dx
chain rule

Question 29

d/dx (xy) =

Answer 29

x dy/dx + y
product rule

Question 30

d/dx (x/y) =

Answer 30

y-x.dy/dx / y^2
quotient rule

Question 31

how do you connect rates of change involving more than 2 variables?

Answer 31

using chain rule to form differential equations

Question 32

forming differential equations