U1 - Derivatives Flashcards
What are the conditions that make a function CONTINUOUS?
- 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
What are the RULES FOR LIMITS?
- 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
Shortcut to find HA (horizontal asymptote).
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
What is a rate? Give an example.
A comparison between two quantities with different units (ex. Speed, concentration)
What is the instantaneous rate of change?
The gradient of the tangent to the graph at a specific point.
What is the average rate of change?
The gradient of the secant between two specific points on a graph.
State the First Principles equation to determine the derivative function.
f’(x) = [f(x+h) - f(x)] / h
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))
f’(a) = lim h>0 [f(a+h) - f(a)] / h
*Any h’s leftover will = 0 and cancel.
What are the derivatives for EXPONENTIAL and LOGARITHMIC functions?
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)
What is the CHAIN RULE?
(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)
State the DIFFERENTIATION RULES.
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)
What is the NORMAL LINE?
The line perpendicular to the tangent of a point on a curve.
State the derivatives of trig functions.
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)
What are the LAWS OF LOGARITHMS?
For a>0, b>0:
ln(ab)=ln a + ln b
ln(a/b)=ln a - ln b
ln(a^n)=n • ln a
State the product rule.
f’(x)=[u’(x)v(x)] + [u(x)v’(x)]
State the quotient rule.
f’(x)=[u’(x)v(x) - u(x)v’(x)] / [v(x)^2]
The IROC and the f’’(x) are respectively known as…
The VELOCITY function and the ACCELERATION function.
Describe the nature of the derivatives as the functions increase and decrease.
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.
What is the rule for rewriting exponents to radicals?
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
What is Leibniz notation for the CHAIN RULE?
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).)
If a question is asking something like, “Find the COORDINATES of…” then __________________ ?
… 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 (+/-)
e^ln = ln e = ___________ ?
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