Chapter 8 - Differentiation and integration Flashcards
Define “limit” (limit of functions)
A function f(x) is said to approach a limit L as x approaches the value a if, given any small positive quantity ß, it is possible to find a positive number K such that |f(x) - 1| < ß for all x satifying 0 < |x - a| < K.
Less formally, this means that we can make the value of f(x) as close as we please to L by taking x sufficiently close to a. Note that, using the formal defnition, there is no need to evaluate f(a); indeed, f(a) may or may not equal L. The limiting value of f as x -> a depends only on nearby values!
What is a continuous function?
If the function f is defined for a and and for values near a, the function f is continuous for x = a and the two equations (picture) are satisfied.
What is the definition of the derivative of the function f(x) at the point x?
What are the conditions for f ‘ (x) to exist in the interval [a, b]?
The function f must be “well behaved”, meaning it has to be continuous, it can’t be vertical, and have no sharp corners.
Define the constant multiplication rule for differentiation.
Define the sum rule of differentiation.
Define the product rule of differentiation.
Define the quotient rule of differentiation.
(ax + b) ‘ = ___
(ax + b) ‘ = a
If f(x) = C, where C is a constant, what is f ‘ (x)?
f ‘ (x) = 0
For all real numbers n, what is (xn)’ ?
(xn)’ = n * xn-1
d/dx (u(x) + v(x)) = (u(x) + v(x)) ‘ = ____
(u(x) + v(x)) ‘ = u’(x) + v’(x)
Solve:
(x2+x3) ‘ = ___
= (x2) ‘ + (x3) ‘ = 2x + 3x2
Using the quotient rule (or the exponent rule), what is (1/x) ‘ ?
(1/x) ‘ = - (1/x2)
Define the chain rule in differentiation.
Use the chain rule to differentiate y = (3x + 1)2
y ‘ = (u(x)) ‘ * u(x) ‘
= 2(3x + 1) * 3
= 6(3x + 1)
= 18x + 6
Solve the equation (picture) using the chain rule.
tan x ‘ ?
d/dx (sin-1x) = (sin-1x) ‘ = ___
d/dx (cos-1x) = (cos-1x) ‘ = ___
d/dx (tan-1x) = (tan-1x) ‘ = ___
