U1 - Derivatives

Question 1

What are the conditions that make a function CONTINUOUS?

Answer 1

• if f(a) exists
• if the limit of f(x) as x > a is defined
• if f(a) = limit of f(x) as x > a

Question 2

What are the RULES FOR LIMITS?

Answer 2

• lim c x>a = c
• lim c f(x) x>a = c lim f(x)
• lim [f(x) +/- g(x)] x>a = lim f(x) x>a +/- lim g(x) x>a
• lim f(x)g(x) x>a = lim f(x) x>a • lim g(x) x>a
• lim f(x)/g(x) x>a = lim f(x) x>a / lim g(x) x>a

Question 3

Shortcut to find HA (horizontal asymptote).

Answer 3

If Deg(N)=Deg(D), HA is the quotient of the leading coefficients of the function.

If Deg(N)Deg(D), there is no HA

Question 4

What is a rate? Give an example.

Answer 4

A comparison between two quantities with different units (ex. Speed, concentration)

Question 5

What is the instantaneous rate of change?

Answer 5

The gradient of the tangent to the graph at a specific point.

Question 6

What is the average rate of change?

Answer 6

The gradient of the secant between two specific points on a graph.

Question 7

State the First Principles equation to determine the derivative function.

Answer 7

f’(x) = [f(x+h) - f(x)] / h

Question 8

State the first principles equation for the gradient of the tangent to y=f(x) at the point where x=a (hint: notation is f’(a))

Answer 8

f’(a) = lim h>0 [f(a+h) - f(a)] / h

*Any h’s leftover will = 0 and cancel.

Question 9

What are the derivatives for EXPONENTIAL and LOGARITHMIC functions?

Answer 9

If y=a^x (a>0), then y’= a^x ln a

If f(x)=e^x, then f’(x)=e^x

If y=ln x, then y’=1/x

If y=ln[f(x)], then y’=f’(x)/f(x) (using CHAIN RULE)

If y=e^f(x), then y’=f’(x)•e^f(x)

Question 10

What is the CHAIN RULE?

Answer 10

(For composite functions)

The derivative of the outside function by the inside function times the derivative of the inside function.

y’ = g’(f(x)) • f’(x)

Question 11

State the DIFFERENTIATION RULES.

Answer 11

if f(x)=c, f’(x)=0

if f(x)=x^n, f’(x)= nx^n-1

if f(x)=c•u(x), f’(x)=c•u’(x)

if f(x)=u(x)+v(x), f’(x)=u’(x)+v’(x)

Question 12

What is the NORMAL LINE?

Answer 12

The line perpendicular to the tangent of a point on a curve.

Question 13

State the derivatives of trig functions.

Answer 13

Non-composite:
If f(x)=sin x, then f’(x)=cos x
If f(x)=cos x, then f’(x)= -sin x
If f(x)=tan x, then f’(x)= sec^2 x
*sec=inverse of cos

Composite (CHAIN RULE)
If y=sin[f(x)], then y’=cos[f(x)]•f’(x)
If y=cos[f(x)], then y’=-sin[f(x)]•f’(x)
If y=tan[f(x)], then y’=sec^2[f(x)]•f’(x)

Question 14

What are the LAWS OF LOGARITHMS?

Answer 14

For a>0, b>0:

ln(ab)=ln a + ln b

ln(a/b)=ln a - ln b

ln(a^n)=n • ln a

Question 15

State the product rule.

Answer 15

f’(x)=[u’(x)v(x)] + [u(x)v’(x)]

Question 16

State the quotient rule.

Answer 16

f’(x)=[u’(x)v(x) - u(x)v’(x)] / [v(x)^2]

Question 17

The IROC and the f’’(x) are respectively known as…

Answer 17

The VELOCITY function and the ACCELERATION function.

Question 18

Describe the nature of the derivatives as the functions increase and decrease.

Answer 18

As the function INCREASES, the slope of f’(x) at (a,b) is POSITIVE.

As the function DECREASES, the slope of f’(x) at (a,b) is NEGATIVE.

Question 19

What is the rule for rewriting exponents to radicals?

Answer 19

If x is rooted, then its exponent form is x^1/2.

In other words, if the radical is: n|x^m, then the exponent is x^m/n

Question 20

What is Leibniz notation for the CHAIN RULE?

Answer 20

dy/dx = dy/du • du/dx

In other words, the derivative of y in terms of u, multiplied by the derivative of u in terms of x.

(The ‘denominator’ indicates the function of which the ‘numerator’ is ‘in terms…’ So if f(x) is made of some term(s) of x, then the derivative of y (dy) is IN TERMS OF x (dx).)

Question 21

If a question is asking something like, “Find the COORDINATES of…” then __________________ ?

Answer 21

… you are likely finding TWO points using an equation from which you can deduce the x-coordinates using the QUADRATIC FORMULA.

You might also have to root two sides of an equation. In that case, remember that when you ROOT a number, you could be getting a NEGATIVE value or a POSITIVE (+/-)

Question 22

e^ln = ln e = ___________ ?

Answer 22

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