trang web hay về toán

https://brilliant.org/practice/joy-of-problem-solving-logical-reasoning/?chapter=introduction-to-joy

http://nm.mathforcollege.com/topics/taylor_series.html: phương pháp số

Tai lieu cho Calculus

https://ptvtp.wordpress.com/tai-li%E1%BB%87u/

Công cụ vẽ hình online

http://www.mathsisfun.com/data/function-grapher.php#functions

http://www.wolframalpha.com/input/?i=graph+x%5E3+-+6x%5E2+%2B+4x+%2B+12

How Polynomials Behave

Source http://www.mathsisfun.com/algebra/polynomials-behave.html

A polynomial looks like this:

example of a polynomial Continuous and Smooth There are two main things about the graphs of Polynomials: The graphs of polynomials are continuous, which is a special term with an exact definition in calculus, but here we will use this simplified definition: we can draw it without lifting our pen from the paper The graphs of polynomials are also smooth. No sharp "corners" or "cusps" How the Curves Behave Let us graph some polynomials to see what happens ... ... and let us start with the simplest form: f(x) = xn Which actually does interesting things: Even values of "n" behave the same: Always above (or equal to) 0 Always go through (0,0), (1,1) and (-1,1) Larger values of n flatten out near 0, and rise more sharply And: Odd values of "n" behave the same Always go from negative x and y to positive x and y Always go through (0,0), (1,1) and (−1,−1) Larger values of n flatten out near 0, and fall/rise more sharply Power Function of Degree n Next, by including a multiplier of a we get what is called a "Power Function": f(x) = axn f(x) equals a times x to the "power" (ie exponent) n The "a" changes it this way: Larger values of a squash the curve (inwards to y-axis) Smaller values of a expand it (away from y-axis) And negative values of a flip it upside down Example: f(x) = ax2 a = 2, 1, ½, −1 Example: f(x) = ax3 a = 2, 1, ½, −1 We can use that knowledge when sketching some polynomials: Example: Make a Sketch of y=1−2x7 Start with the simplest "odd power" graph of x3, and gradually turn it into 1−2x7 We know how x3 looks, x7 is similar, but flatter near zero, and steeper elsewhere, Squash it to get 2x7, Flip it to get −2x7, and Raise it by 1 to get 1−2x7. Like this: So by doing this step-by-step we can get a good result. Turning Points A Turning Point is an x-value where a local maximum or local minimum happens: How many turning points does a polynomial have? Never more than the Degree minus 1 The Degree of a Polynomial with one variable is the largest exponent of that variable. Example: a polynomial of Degree 4 will have 3 turning points or less x4−2x2+xhas 3 turning points x4−2xhas only 1 turning point The most is 3, but there can be less. We may not know where they are, but at least we know the most there can be! What Happens at the Ends And when we move far from zero: far to the right (large values of x), or far to the left (large negative values of x) then the graph starts to resemble the graph of y = axn where axn is the term with the highest degree. Example: f(x) = 3x3−4x2+x Far to the left or right, the graph will look like 3x3 Near Zero, they are different Far From Zero, they become similar This makes sense, because when x is large, then x3 is much greater than x2 etc This is officially called the "End Behavior Model". And yes, we have come to the end! Summary Graphs are continuous and smooth Even exponents behave the same: above (or equal to) 0; go through (0,0), (1,1) and (−1,1); larger values of n flatten out near 0, and rise more sharply. Odd exponents behave the same: go from negative x and y to positive x andy; go through (0,0), (1,1) and (−1,−1); larger values of n flatten out near 0, and fall/rise more sharply Factors: Larger values squash the curve (inwards to y-axis) Smaller values expand it (away from y-axis) And negative values flip it upside down Turning points: there are "Degree − 1" or less. End Behavior: use the term with the largest exponent Question 1 Functions (Calculus, General) Help Which one of the following could be the function for the above graph? A f(x) = 3 - 2x5 B f(x) = 3 - 2x6 C f(x) = 3 + 2x6 D f(x) = 3 - 2x2

The graph is symmetrical about the y-axis, which means the function is even. This rules out answer A.
The graph is upside down indicating that the power of x should be preceded by a negative. This means the answer could be B or D.
Since the curve is flattened at the top and does not show the characteristic shape of a parabola, we conclude that B is the correct answer.
Furthermore, the y-intercept is 3 which agrees with this answer.

Question 3

Functions (Calculus, General)

Help

The above shows the graph of y = 2x⁵- 6x³
How many turning points does it have?

A

2

B

3

C

4

D

Cannot say

+0.50

Excellent ... you are right.

At a maximum turning point the slope of the curve changes from positive to zero to 

negative - there is a maximum turning point at approximately (-1.4, 5.8).

At a minimum turning point the slope of the curve changes from negative to zero to

 positive - there is a minimum turning point at approximately (1.4, -5.8).

At the origin the slope of the curve changes from negative to zero and then back to 

negative - this is called a point of inflection and is not a turning point.

Far away from the origin the graph of y = 2x⁵-6x³ behaves like the graph of y = 2x⁵,

 so there will be no further turning points.

Therefore, there are just 2 turning points.

Question 5

Functions (Calculus, General)

Help

Which one of the following could be the function for the above graph?

A

f(x) = 1 - 3x⁴

B

f(x) = 1 + 3x⁴

C

f(x) = (1 - 3x)⁴

D

f(x) = (1 + 3x)⁴

+0.50

Excellent ... you are right.

The graph is symmetrical about the y-axis, which means the function is even.

When expanded the answers C and D have both even and odd powers, so they are
neither even nor odd functions. So the answer cannot be C or D.

Which leaves only A or B.

The graph is upside down indicating that the power of x should be preceded by a negative.
Therefore the answer is A.

Question 6

Functions (Calculus, General)

Help

Which one of the following could be the function for the above graph?

A

y = x⁶ - 5x⁴ + 7x²

B

y = x³ + 7x

C

y = x⁵ - 5x³ + 7x

D

y = x⁵ - 5x² + 7x

+0.50

Yes! Well Done.

The graph has point symmetry about the origin, which means the function is odd.

 This rules out answer A which is even and answer D, which is neither even nor odd.
The graph has 4 turning points, so it cannot be a cubic - a cubic has at most 3 - 1 = 2 

turning points. So answer B cannot be correct.
A quintic, on the other hand, has at most 5 - 1 = 4 turning points, so C could be the

 correct answer

Question 8

Functions (Calculus, Hard)

Help

Use the Function Grapher at http://www.mathsisfun.com/data/function-grapher.php to find an estimate, correct to 1 decimal place, of the minimum value of the function
f(x) = 5x⁴ - 3x⁴ + 2x

(Note: enter function as 5x^4-3x^4+2x ... or in a simpler form if you can think how.)

A

-0.6

B

-0.8

C

-0.9

D

-1.9

+0.50

Congratulations, that is the right answer.

Using the Function Grapher, my first attempt to the solution is that it has a minimum point somewhere between x = -1 and x = 0.


When I zoom in, I get a better approximation, which is the point (-0.6, -0.9) to 1 decimal place.

So the minimum value of the function is -0.9 correct to 1 decimal place.

(Note: the function could have been simplified to f(x) = 2x⁴ + 2x)

Intermediate Value Theorem

Source http://www.mathsisfun.com/algebra/intermediate-value-theorem.html

The idea behind the Intermediate Value Theorem is this:

When we have two points connected by a continuous curve: one point below the line the other point above the line ... then there will be at least one place where the continuous curve crosses the line!

Well of course we must cross the line to get from A to B!

Now that you know the idea, let's look more closely at the details.

Continuous

The curve must be continuous ... no gaps or jumps in it.

Continuous is a special term with an exact definition in calculus, but here we will use this simplified definition:

we can draw it without lifting our pen from the paper

More Formal

Here is that idea stated more formally:

		When: The curve is the function y = f(x), which is continuous on the interval [a, b], and w is a number between f(a) and f(b),
		Then ...

... there must be at least one value c within [a, b] such that f(c) = w

In other words the function y = f(x) at some point must be w = f(c)

Notice that:

• w is between f(a) and f(b), which leads to ...
• c must be between a and b

At Least One It also says "at least one value c", which means we could have more. Here, for example, are 3 points where f(x)=w.

How Is This Useful?

Whenever we can show that:

• there is a point above a line
• and a point below a line, and
• that the curve is continuous,

we can then safely say "yes, there is a value somewhere in between that is on the line".

Example: is there a solution to x⁵ - 2x³ - 2 = 0 between x=0 and x=2?

At x=0:

0⁵ - 2 × 0³ - 2 = -2

At x=2:

2⁵ - 2 × 2³ - 2 = 14

Now we know:

• at x=0, the curve is below zero
• at x=2, the curve is above zero

And, being a polynomial, the curve will be continuous,

so somewhere in between, the curve must cross through y=0

Yes, there is a solution to x⁵ - 2x³ - 2 = 0 in the interval [0, 2]

Question 1

Functions (Calculus, General)

Help

Between which of the following two values does the equation 3x³ + 5x - 11 = 0 

have a solution?

A

Between -2 and -1

B

Between -1 and 0

C

Between 0 and 1

D

Between 1 and 2

D is the correct answer

Let f(x) = 3x³ + 5x - 11
Therefore
f(-2) = 3 × (-2)³ + 5 × (-2) - 11 = -24 - 10 - 11 = -45 < 0
f(-1) = 3 × (-1)³ + 5 × (-1) - 11 = -3 - 5 - 11 = -19 < 0
f(0) = 3 × 0³ + 5 × 0 - 11 = 0 - 0 - 11 = -11 < 0
f(1) = 3 × 1³ + 5 × 1 - 11 = 3 + 5 - 11 = -3 < 0
f(2) = 3 × 2³ + 5 × 2 - 11 = 24 + 10 - 11 = 23 > 0

Since f(1) < 0 and f(2) > 0 and f is a polynomial, the Intermediate Value Theorem tells us that f(x) = 0 for some value of x between 1 and 2.

Question 3

Functions (Calculus, General)

Help

Between which of the following two values does the equation 

-7x³ + 20x² - x + 1 = 0 have a solution?

A

-1 and 0

B

0 and 1

C

1 and 2

D

2 and 3