Calculus 3.6 and 3.7

Keywords and definitions for NCEA Level 3 calculus, 3.6 (Differentiation) and 3.7 (Integration)

f(x)= ln(x)

f(x)= ln(x)

f’(x)= 1/x

f(x)= e^ax

f’(x)= ae^ax

f’(x)=e^ax

f(x)=e^ax/a

f(x)= sin(x)

f’(x)= cos(x)

f(x)= cos(x)

f’(x)= -sin(x)

f(x)= tan(x)

f’(x)= sec²(x)

f(x)= sec(x)

f’(x)= sec(x)tan(x)

f(x)= cosec(x)

f’(x)= -cosec(x)cot(x)

f(x)= cot(x)

f’(x)= -cosec²(x)

Product Rule

y’=uv’+u’v

Quotient Rule

y’= (u’v-v’u)/v²

Chain Rule

(u(v))’=u’(v)v’

Concave up

f’(x) is increasing

Concave down

f’(x) is decreasing

Stationary point

f’(x)=0

Local minimum

f’’(x)>0

Local maximum

f’’(x)<0

Area between two curves

[(lower curve)’-(upper curve)’]

Tangent line

Straight line with same gradient as a point.

Normal line

Straight line perpendicular to the tangent line.

s(t)

displacement

v(t)

velocity

a(t)

acceleration

Continuous

A function is one continuous line, it can be drawn without lifting the pen

Differentiable

A function is continuous, and with no corners or sharp points (it is all one function)

c

Constant of integration

Area under curves bound by x-axis

Integrate with respect to y

Area under curves bound by y-axis

Integrate with respect to x

Calculate distance

Area under displacement graph

Finding gradient

Differentiate then substitute in known point

Finding equation of tangent

Find (x,y) using original function. Differentiate and substitute x to find gradient. Use y-y1=m(x-x1) to find equation

Finding equation of normal

Find (x,y) using original function. Differentiate and substitute to find tangent gradient. Divide this by -1 to find normal gradient. Use y-y1=m(x-x1) to find equation

Integration by substitution

Substitute u=g(x), find dx=du/u. If needed, find x=g(u). Substitute and integrate. Substitute g(x) back in for u.

Differentiate parametric function

dy/dx=dy/dt * dt/dx

Cancel out e

ln both sides

Cancel out ln

e^ both sides

Integrating differential equations

Get wanted variable on one side, integrate both sides. Simplify wanted variable using e. Substitute in known values to solve for unknown. Then substitute to find answer.

Differentiation involving geometric shapes

Find known derivative and write as a function of known functions (e.g. dV/dt=dV/dr*dr/dt). Solve.

Area above x-axis

Positive (add together)

Area below x-axis

Negative (subtract from)

Average area under sine/cosine graph

0

Secant line

Straight line connecting two points on a curve

Exponential growth/decay

y=Ae^kt