Week 4-5 Flashcards
(27 cards)
What is the slope of a tangent to a curve?
The slope of the tangent to a curve at this point is known as the derivative of the function with respect to x
What are the three ways to represent the slope of a tangent to a curve?
m = (y2 - y1)/(x2 - x1)
m = ( f(x) - f(a) )/h
m = f’(a) = lim(h–>0) ( f(x + h) - f(a) )/h
What is a differentiable function?
A function f(x) is differentiable at “a” is f’(a) exists. It is differentiable on an open interval (a,b) if it exists at any point on the interval.
The tangent line to y = f(x) at (a,f(a)) is the line with slope equal to…
y - f(a) = f’(a)(x-a)
in the general form
yo = f(a), xo = a, f’(a) = m
y - f(a) = m(x-a)
If f(a) is differentiable at “a”, does it mean f(a) is continuous at “a”? Is the opposite true?
Yes, if f(a) is differentiable at a, then f is continuous at a. Differentiability implies continuity because for f to be differentiable at a, the limit that defines the derivative must exist, which in turn requires that f(x) approaches f(a) as x approaches a.
However, the opposite is not necessarily true: a function can be continuous at a point but not differentiable there. A common example is the absolute value function, f(x) = IxI, but not differentiable at that point because it has a sharp corner.
How is the interval of ln(sin(x^2)) restricted?
It is well defined on the positive values as x^2 is always postive.
Define composition function
Consider the functions f(x) and g(x), and suppose that the image of g(x) is in the domain of f(x), then the composition of the functions f,g is a new function such that
h(x) = f ∘ g = f(g(x))
How is the interval restricted on lin(sin(x^2))?
It is well defined for positive values of sin(x^2) since the domain is of ln
Define the chain rule
Consider the functions f(x), g(x), where the image of g(x) is the domain of f(x). Suppose that f,g are differentiable in their domains, then the derivative of the composition f(g(x)) is
(f(g(x))’ = f’(g(x))g’(x)
Can the chain rule be combined with all differentiation rules?
Yes
Is the chain rule applied to all possible layers?
Yes
Can derivatives be applied successively?
Yes
What are the notations for the first, second, third and nth derivative?
f’(x), f’‘(x), f’’‘(x) and f^n(x)
When is a function one-to-one or inyective?
A function is one-to-one or inyective if for every x1, x2 in the domain of f(x) we have x1 ≠ x2 then f(x1) ≠ f(x2)
How do we verify if a function is one-to-one?
We use the horizontal line test to verify if a function is one-to-one. If the line crosses the function only at a point we say that the function is one-to-one.
The image of f is the domain of what? The domain of f is the image of what?
Domain of f^-1, Image of f^-1
The inverse function is found by writing…
y = f(x)
How do you obtain the inverse trigonometric function?
To obtain the inverse trigonometric function, first we have to restrict the domain so that the function is one-to-one. This is because they are period and so do not have inverses otherwise.
What is the formula for the derivative of an inverse function?
f(f^-1(x)) = x
(f^-1(x))’ = 1/f^-1(f^-1(x))
d/dx(sin^-1(x))
1/sqrt(1-x^2)
d/dx(cos^-1(x))
-1/sqrt(1-x^2)
d/dx(tan^-1(x))
1/(1+x^2)
d/dx(csc^-1(x))
-1/(x(sqrt(x^2 - 1))
d/dx(sec^-1(x))
1/(xsqrt(x^2 - 1))
