Minimum point if d2f/dx2 is...? Maximum point if d2f/dx2 is...?

Minimum point if d2f/dx2 is \> 0 Maximum point if d2f/dx2 is \<0

Dr Griffiths Flashcards by Steven Baird

Define a function

A function is a rule which operates on an input and produces a single output

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What do you do to the number of a;

“One-to-one” rule?

“Many to one”?

“One to many”?

“One-to-one” - Multiply by 1

“Many to one” - Square the numbers

“One to many” - Take the square root

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“One to many” is not an exmaple of a ____

Why?

Function

Because one input produces more than one output

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A function has a ____ and ____

domain** and **range

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Define domain

A set of values we allow the independant variable to take

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Define the range

The set of y-values in the domain

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Functions can also be described…?

Define it

Parametrically : One of a set of independent variables that express the coordinates of a point

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Real life example of continuity?

Can you keep your pen on the paper when drawing the curve?

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A function is ____ if it is not continuous

discontinuous

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In the unit step function,

v(t) = ? if t ≥0

v(t) = 0 if t ? 0

v(t) = 1 if t ≥0

v(t) = 0 if t < 0

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A function that has a definite pattern repeated at regular intervals is said to be…?

e.g.?

Periodic

e.g. y=sinx

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A function f(x) is ____ if we can find a number ‘T’ such that

f(x+t) = ? for all x

A function f(x) is periodic if we can find a number ‘T’ such that

f(x+t) = f(x) for all x

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A function that is symetric about the _-axis is said to be…?

y-axis is said to be even

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A even function is such that

f(-x) = ? for all of x

f(-x) = f(x) for all of x

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A odd function is such that

f(-x) = ? for all of x

f(-x) = -f(x) for all of x

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An exponential function has the form y = ?

y = a^x

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y = 2^x - derivative lies ____ the curve

y = 3^x - derivative lies ____ the curve

y = 2^x - derivative lies below the curve

y = 3^x - derivative lies above the curve

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3! = ?

3! = 3x2x1 = 6

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e^x - exponential ____ as x → ?

e^-^x- expontential ____ as x → ?

e^x - exponential growth as x → ∞

e^-^x- expontential decay as x → ∞

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e always…?

dominates

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If y=e^x then x = ?

x = ln(y)

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If it’s e^x + e^x then it’s ?

But if its e^x - e^-^xthen it’s ?

Even

Odd

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tanh(x) = ?

coth(x) = ?

sech(x) = ?

cosech(x) = ?

tanh(x) = sinh(x)/cosh(x)

coth(x) = 1/tanh(x)

sech(x) = 1/cosh(x)

cosech(x) = 1/sinh(x)

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The three different types of stationary points are…?

Maximum point
Minimum point
Points of inflection

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Maximum and minimum points can either be ____ or \_\_\_\_

**_local**_ or _**global_**

**Minimum** point if d²f/dx² is...? **Maximum** point if d²f/dx²is...?

**Minimum** point if d²f/dx² is \> 0 **Maximum** point if d²f/dx²is \<0

If d²f/dx²= 0 ...? so we need to?

we need more information. So we need to evaluate the sign of the function f(x) near to the point x=x_s

Any change in sign reveals a...?

point of inflection

The **Maclaurin series** always expands about the point...?

x=0

**Maclaurin series** = **Taylor series** about...?

**x=0**

e = ?

e = 1 + 1/1! + 1/2! + 1/3! + ... 1/n!

The limit of f(x) is _ as x approaches a, and is written as...?

The limit of f(x) is L as x approaches. This is written as lim_x→a f(x) = L

L'Hôpital's Rule, if f(x)/g(x) = 0/0 you do what?

Just differentiate both lines f'(x)/g'(x)

A sequence is simply...?

...a sucession of numbers

A series is...?

...simply the sum of the terms of the sequence

If the sequence of partial sums converges then we say that...??

...the series converges

What would be the first thing to spot in something like this? I = ∫ sin(x) cos(x) dx

That if you dy/dx sin(x) then you get cos(x) so **d(sin(x))/dx** = cos (x) so sub in the bold part into the previous equation for cos(x)

Tricky intergrates can often be simplified by...?

...using a suitable substitution

The mean value of a function f(x) between points A and B is...?

The root means square value of a function f(x) between points A and B is...?

The RMS (\_\_\_\_ ____ \_\_\_\_) value is used when...?

The RMS (**_Root mean square_)** value is used when... **determining the strength of voltage supplies**

What is the area of a circle?

πr²

A solid rotating about the x-axis, r = ? Equation is v =

r=y v =

Rotating about the y-axis, r = ? Equation v = ?

r = x v =

How do you measure the length of a curve? (Eqn)

Simply apply pythagoras' theorem to determine the length of a hypotenuse (kinda)

An **explicit** function has the form y = ? Whereas an **implicit** function has the form...

**Explicit** y = f(x) **Implicit** f(x,y) = 0

y = arcsin(x) What does x = ?

x = sin(y)

We can use the idea of implicit differentiation to determine the...

...derivative of logarithmic type functions

Define **Partial differentiation**

**Partitial differentiation** is define as the rate of change of a function of multiple variables, with respect to one (or more) independant variables

∂ = ? d = ?

∂ = Many D = One

When taking the partial derivative with respect to one independant variable the others are viewed as...?

...constants

f_xx means?

Differentiate with respect to x twice

Stationary points occur when...?

df/dx = 0

What would functions of two variables look like?

= f_xxf_yy - f(xy)²

If D \< 0 = ? If D \> 0 and f_xx \> 0 = ? If D \> 0 and f_xx \< 0 = ? If D = 0

If D \< 0 = **Saddle point** If D \> 0 and f_xx \> 0 = **Minimum point** If D \> 0 and f_xx \< 0 = **Maximum point** If D = 0 - **We need some more info**

D = ? f(xy) = ?

D = f_xxf_yy - f(xy)² f(xy) = 0 (for whatever reason)

j = ?

√-1

j is the...

...'Imaginary number'

In general, we write complex numbers in the form;

z = 1 +- j

The modulus of a complex number is...?

|z| = √(a²+b²)

Give an example of a complex conjugate pair

z = a + bj z\* = a - bj

What do you plot complex numbers on?

An **Argand diagram**

On an **Argand diagram**, what is the x and y-axis?

x-axis = **real part of z** y-axis = **imaginary part of z**

For complex numbers, x = ? y = ?

x = **rcos(ø)** y = **rsin(ø)**

In an **argand diagram** when plotting complex numbers, what parts of the diagram do you add or minus π?

3rd = -π 2nd = +π ø \> π other way round

What is **Elver's identity?**

e^jπ + 1 = 0

re^jø = ?

re^jø = r(cos(ø) + jsin(ø))

What is **De Moivre's theorem?**

cos(nø) + jsin(nø) = [cos(ø) + jsin(ø)]ⁿ

In **De Moivre's theorem,** cos(nø) + jsin(nø) = ?

cos(nø) + jsin(nø) = [cos(ø) + jsin(ø)]ⁿ

**De Moivre's theorem** is particulary useful for...?

...finding roots

Non linear homogeneous equations = ? Linear non-homegneous = ?

NLH = 0 LNH = x²

The function I(x) is known as...?

The **intergrating factor**

For first ODE's, what do we want it to look like?

dy/dx + yp(x) = q(x)

I(x) = ?

In first ODE's, v(x) = ?

Dr Griffiths Flashcards

(76 cards)