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 2The 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 3The 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 4The 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?

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

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?

Re-express as a log which is log [0.035] 2.74 = x

1. Draw line
2. log 2.74__ = x
   log 0.035

Or is this correct?
0.035 [x] = 2.74 Find x?

1. Log both sides
2. log 0.035 [x] = log 2.74
3. Re-express to remove power
   x log 0.035 = log 2.74
4. Make x the subject of the equation
   x = log 2.74 / log 0.035

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?

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?

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 thanvolves some aspect of Maths, give some real life scenarios that involves the exponent aspect of maths.

A curve graph

What is the difference between a straight line and a curve graph?

Can one graph consist of a straight line graph and a curve graph?

If yes, can they ever intersect?

How can one determine by calculation if these two graphs intersect?

Intersection of two straight lines

Can a graph consist of more than one straight lines?

If yes, why would someone put more than one straight line graphs on the same graph?

In the event that the straight lines intersect, what does this signify?

How can one calculate that point to verify that the graph is accurate?

Gradient and intercepts

How do you find the gradient of a straight line?

What is the use of finding gradient?

How can you explain gradient so that the ratio of change in y over change in x is not mixed up?

What is the purpose of finding y-intercept?

How do find the y-intercept?

What is the meaning of intercept?

A graph commonly consists of two axes called the x-axis (horizontal) and y-axis (vertical). Each axis corresponds to one variable. This variable represent something that a human can relate to for example the axes can be labelled with different names, such as velocity, time, height, temperature, price or quantity.
The place where the two axes intersect is called the origin. The origin is also identified as the point (0,0). Parts of a graph
- x-axis

- y-axis

- origin

A point on a graph represents a relationship. Each point is defined by a pair of numbers containing two co-ordinates (x and y). A co-ordinate is one of a set of numbers used to identify the location of a point on a graph. The co-ordinate is measured from the origin. First the x co-ordinate is given which states movement to the left and to the right. Second the y co-ordinate is given which states movement up and down.

Rate of change

What is meant by change in x-axis

What is meant by change in y-axis

What is meant by change in y-axis to change in x-axis?

What does the change in distance w.r.t. time signify?

How is this depicted or represented on a graph paper?

What is the gradient?