Week 4: Differentiation, hyperbolic functions, Newton's method Flashcards
What is the product rule?
If
f(x) = uv
then
f’(x) = y’ = uv’ +u’v
What is the quotient rule?
If
f(x) = u/v
then
f’(x) = y’ = (vu’ - uv’) / (v^2)
What is the chain rule?
dy/dx = (dy/du) x (du/dx)
What is an explicit function?
An explicit function has
x as an independent variable
and
y as a dependant variable
What is an implicit function or relationship?
A relationship were it cannot be rearranged to make the equation explicit (y in terms of x).
What is the test for an odd function?
Test for odd function:
f(-x) = -f(x)
What is the test for an even function?
Test for an even function:
f(x) = f(-x)
What is the method for implicit differentiation?
- Differentiate both sides with respect to x
2. Rearrange so that dy/dx is the subject of the equation
What is Newton’s Method?
Newton’s Method is an iterative process, used to approximate the solutions to an equation.
What form must an equation be in to apply Newton’s Method?
Equation must be in the form
f(x) = 0
What is the standard form of Newton’s Method?
Standard form of Newton’s method is
x1 = x0 * (f(x0)/f’(x0))
Under what conditions will Newton’s method fail?
Newton’s method will fail if f’(x0) = 0.
As under no circumstances can you divide by 0.
What is the hyperbolic cosine (coshx) equal to?
coshx = 1/2 (e^x + e^(-x))
What is the hyperblic sine (sinhx) equal to?
sinhx = 1/2(e^x - e^(-x)
What is the hyperbolic tangent (tanhx) equal to?
tanhx = sinhx/coshx
What is the hyperbolic secant (sechx) equal to?
sechx = 1/coshx
What is the hyperbolic cosecant (cosechx) equal to?
cosechx = 1/sinhx
What is the hyperbolic cotangent (cothx) equal to?
cothx = 1/tanhx
What are the properties of the hyperbolic cosine (coshx) graph? Describe the shape of the graph.
Domain: all real x
Range: 1 ≤ y ≤ ∞
Even function: cosh(x) = cosh(-x)
The graph is a similar shape to a parabola. The base graph (unchanged) has no x-intercepts and a single y-intercept at y =1.
What are the properties of the hyperbolic sine (sinhx) graph? Describe the shape of the graph.
Domain: All x
Range: All y
Odd function
sinh(-x) = -sinh(x)
The graph has a similar shape to that of a cubic. The base graph (unchanged) passes through the origin (0,0).
What are the properties of the hyperbolic tangent (tanhx) graph? Describe the shape of the graph.
Domain: all x
Range: 1 < y < -1
The graph has a similar shape to that of inverse tan. The graph passes through the origin.
The graph has 2 asymptotes (y = 1 and y = -1)
As x ➡ ∞, y ➡ 1
As x ➡ -∞, y ➡ -1
