Are these solutions correct?

Solution 1

y = log base 3 (8x^7 + x)
u = 8 x^7 + x
dy/dx = 56 x^6 + 1
= (56 x^6 + 1) / (8 x^7 + x) log base 3 e

Solution 2
P = ln 6s^3 + s
u = 6s^3
dy/dx = 18 s^2
= (18s^2)/(6s^3) + 1

Solution 2
P = ln 6s^3 + s
u = 6s^3 + s
dy/dx = 18 s^2 + 1
= (18s^2 + 1)/(6s^3 + s)

Solution 3
y = 15 x^3 + 7/(2x^3) - x
y = 15x3 + 7 *2 x^-3 - x
dy/dx = 45 x^2 + 42 x^-4 - 1

Solution 3
y = 15 x^3 + 7/(2x^3) - x
dy/dx = 45 x^2 + 7/6 x^-2 - 1

Log checking

Problem 1
log (2x^2 + 6x) - 6 = log (2x)

What can I do now?


Problem 2
Am I going correct?

log (x + 6) = 2 - log (4x)
log (x + 6) = log 100 - log (4x)

Application of simple differentiation

Question 1
Water is being drained from a pond such that the volume V (in m^3) of water in the pond after t hours is given by V = 5000(60 - t)^2. Find the rate at which the pond is being drained after 4 h.

Question 2
The velocity of an object moving with constant acceleration can be found from the equation v = (v[0] ^2 + 2as)^2, where v[0] is the initial velocity, a is the acceleration, and s is the distance traveled. Find dv/ds.

Example 3
The electric field E at a distance r from a point charge is E = k/r^2, where k is a constant. Find an expression for the instantaneous rate of change of the electric field with respect to r.

Example 4
The distance s (in m) traveled by a subway train after the brakes are applied is given by s = 20t - 2t^2. How far does it travel, after the brakes are applied, in coming to a stop?

Examining the accuracy of logs

Look at the following and made some comments to assist. Remember the log base is in [] brackets

Example 1
log [8] (x + 2) = 2 - log [8] 2
log [8] 1 + log [8] 2 = 2 / (x + 2)

Example 2
log [8] (x + 2) = 2 - log [8] 2
log [8] (x + 2) + log [8] 2 = 2
log (2x + 4) = 2
log (2x + 8) = 2
log 8
2x + 4 = 1.8
2x = 1.8 - 4

Example 3
log [8] (x + 2) = 2 - log [8] 2
log [8] x + log [8] 2 = 2 - log [8] 2
log [8] x + 2 log [8] 2 = 2

Example 4
log [8] (x + 2) = 2 - log [8] 2
log [8] (x + 2) = 1.67
log [8] 1.67 = x + 2

Example 5
log [8] (x + 2) = 2 - log [8] 2
log [8] (x + 2) + log [8] 2 = 2
log [8] 2(x + 2) = 2
log [8] (2x + 4) = 2
log [8] 2x = 2 - 4

Example 6
log [8] (x + 2) = 2 - log [8] 2
log (x + 2) / log 8 = 2 - log 2 / log 8
log (x + 2) / 0.903 = 2 - 0.33
log (x + 2) 0.903 * 1.667
log x + 2 = 1.5
log x = 1.5 - 2
log x = 0.5
x = 0.32

The use of ( )

What is the use of a bracket?
4 (x + 3)
4 (x + 3) (x + 3)
4 (x + 3)squared

When can you open out a bracket and when is it not allowed?

What is the differences in the following?
log 4
log 4x
log 4x + 3
log (4x + 3)
log (4x + 3)squared

Simple Algebraic Expressions

What will you say to students who do the following in an attempt to help them?


Example 1
1.14 x + 3.42 = 2x - 1
2x - 1.14x - 1 - 3.42 = 0

Example 2
1.338x - 0.699 = 0.778x + 2.334
1.338x + 0.778x = 2.334 - 0.699

Exponents

Below show some exponent approaches. This time the power will be in brackets i.e. ( )

Example 1
5 (2x - 1) = 6 (x + 3)
Multiply left by 6
Multiply right by 5
30 (2x - 1) = 30 (x + 3)
2x -1 = x + 3

Example 2
5 (2x - 1) = 6 (x +3)
log (5) 2x - 1 = log (6) x + 3
log 2x - 1 / log 5 = log x + 3 / log 6
12x - 6 = 5x + 15

Example 3
5 (2x - 1) = 6 (x +3)
log (5) 2x - 1 = log (6) x + 3
2x - 1 log 5 = x + 3 log 6
2x - 1 log 5 - x + 3 log 6 = 0
log [5] (2x - 1) + log [6] (-x + 3) = 0
log 10x - 5 + 9 - x = 0
log 9x + 9 = 0
log 9x = 9

Evaluation of simple logs

Examine the following statements. Remember that the base will be represented in (). Help these students.

Example 1
log (4) x = 12
log x = log 12
log 4
log x = 1.08 * 0.6
log x = 0.65
x = 4.5

Example 2
log (4) x = 12
log 4 = 12
x = 12 / log 4
x = 20

Example 3
log (4) x = 12
log (4) x = log 12

Example 4
log (4) x = 12
log (4)12 = x

Example 5
log (8) 32 = x
x = 8/32
x = = 1/4

The [] is used since subscript and superscript are not allowed.
2 [3] = 8 can also be expressed as log [2] 8 = 3


Is this correct?
0.035 [x] = 2.74 Find x?

1. Re-express as a log which is log [0.035] 2.74 = x
2. Draw line
3. log 2.74__ = x

log 0.035

Or is this correct?

0.035 [x] = 2.74 Find x?

1. Log both sides
   log 0.035 [x] = log 2.74
2. Re-express to remove power

x log 0.035 = log 2.74

1. Make x the subject of the equation

x = log 2.74 / log 0.035

Derivative

What really is first derivative?
Is first derivative any thing to do with differentation? Explain.
What really is the use of firat derivative in maths?
How can I understand first derivative in relation to the world?

Limits

What is Limits

Why is limits needed in maths

What is the purpose of limits with respect to graphs

Why is limits used in the delta process

Delta process

gradient = change in y
change in x

why then do I see gradient = lim change in y
∆h→0 change in x

What is the purpose of lim
∆h→0

Why is (x1, y1) always (x, y)

Why for (x2, y2) , x2 is always x + ∆h
What then is y2

In class we refer to a policeman, what is this policeman all about?

What is the denominator always and why?

What is the numerator always and why?

Why the numerator has to factorise to take out ∆h?

How can someone say where ∆h is substitute 0?

Logs

What really is logs?
Do you think this is an important aspect of maths?
How is logs related to exp?
What is the basic strategy in logs?

Missing students

A lot of students have not sign in and hence no comments from you.

Should I just give 0 / 10
You are to comment on each post.
If your comment is none of the relevant types, it should be viewed as none.

Next week I will post the student names or ids

Exponent

What is the purpose of exponent?
Is a quadratic an exponent?
It is said that everything in life involves some aspect of Maths, give some real life scenarios that involves the exponent aspect of mahs.