Week 4: Differentiation, hyperbolic functions, Newton's method

Question 1

What is the product rule?

Answer 1

If
f(x) = uv
then
f’(x) = y’ = uv’ +u’v

Question 2

What is the quotient rule?

Answer 2

If
f(x) = u/v
then
f’(x) = y’ = (vu’ - uv’) / (v^2)

Question 3

What is the chain rule?

Answer 3

dy/dx = (dy/du) x (du/dx)

Question 4

What is an explicit function?

Answer 4

An explicit function has
x as an independent variable
and
y as a dependant variable

Question 5

What is an implicit function or relationship?

Answer 5

A relationship were it cannot be rearranged to make the equation explicit (y in terms of x).

Question 6

What is the test for an odd function?

Answer 6

Test for odd function:

f(-x) = -f(x)

Question 7

What is the test for an even function?

Answer 7

Test for an even function:

f(x) = f(-x)

Question 8

What is the method for implicit differentiation?

Answer 8

1. Differentiate both sides with respect to x

2. Rearrange so that dy/dx is the subject of the equation

Question 9

What is Newton’s Method?

Answer 9

Newton’s Method is an iterative process, used to approximate the solutions to an equation.

Question 10

What form must an equation be in to apply Newton’s Method?

Answer 10

Equation must be in the form

f(x) = 0

Question 11

What is the standard form of Newton’s Method?

Answer 11

Standard form of Newton’s method is

x1 = x0 * (f(x0)/f’(x0))

Question 12

Under what conditions will Newton’s method fail?

Answer 12

Newton’s method will fail if f’(x0) = 0.

As under no circumstances can you divide by 0.

Question 13

What is the hyperbolic cosine (coshx) equal to?

Answer 13

coshx = 1/2 (e^x + e^(-x))

Question 14

What is the hyperblic sine (sinhx) equal to?

Answer 14

sinhx = 1/2(e^x - e^(-x)

Question 15

What is the hyperbolic tangent (tanhx) equal to?

Answer 15

tanhx = sinhx/coshx

Question 16

What is the hyperbolic secant (sechx) equal to?

Answer 16

sechx = 1/coshx

Question 17

What is the hyperbolic cosecant (cosechx) equal to?

Answer 17

cosechx = 1/sinhx

Question 18

What is the hyperbolic cotangent (cothx) equal to?

Answer 18

cothx = 1/tanhx

Question 19

What are the properties of the hyperbolic cosine (coshx) graph? Describe the shape of the graph.

Answer 19

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.

Question 20

What are the properties of the hyperbolic sine (sinhx) graph? Describe the shape of the graph.

Answer 20

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).

Question 21

What are the properties of the hyperbolic tangent (tanhx) graph? Describe the shape of the graph.

Answer 21

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