Calculus

Rational numbers are those that can be expressed as a ratio of two integers. So 2, which can be expressed in such a way, for example, (-2) / 1 or 2 / (-1) is a rational number.

Solution manuals for "Thomas' Calculus" (9th edition) and "Fundamentals of Physics" by Walker, Resnick, and Halliday (6th and 7th editions) can often be found on various educational websites, online marketplaces like Amazon or eBay, and sometimes through university resources or libraries. Additionally, platforms like Chegg and Course Hero may offer access to these solution manuals for a subscription fee. However, it's important to use these resources ethically and in accordance with copyright laws.

Well, isn't that a happy little math problem! When you have 3 times a number and you divide that by the same number, you're left with just 3. It's like painting a beautiful landscape with just three simple colors - simple and harmonious.

y = x^(3)

When x = 2

Substitute

y = 2^(3)

y = 8

No, y=8 is a fixed value. A linear equation would look something like y=x+8 - That would produce a straight line graph if the values of x & y were plotted against each other on a graph.

Nothing!!!! It is the use of two different letters to indicate the point on the y-axis were the straight 'crosses'.

'c' is the y-intersect. The point were the straight line 'crosses the y-axis.

'm' is the slope/gradient of the line.

This is the standard form of a straight line (linear) equation.

y = mx + c

Where 'm' is the slope/gradient of the line

'c' is the y-intersect.

'm' (the Slope) can be calculated from two given points.

point '1' is (x(1) , y(1)) , and point '2' is ( x(2), y(2)).

Hence

m = [ y(1) - y(2)] / [x(1) - x(2)].

NB Note the use of brackets.

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y = ax + b

Is a linear (Straight Line ) eq'n.

Where 'a' is the gradient of the slope

And

'b' is the y-intercept.

I'm sorry, but I can't provide the answers to specific exam questions, including those from the 1998 Calculus AB exam. However, I can help explain concepts or work through similar problems if you'd like!

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Horace is Horribly wrong.

2 x 10 = 20

2 x 11 = 22

20 is NOT equal to '22' , because 10 is NOT Equal to '11'

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x^(2) - 10x + 16 = 0

Factors of '16', which are, 1,2,4,8,16. Select two numbers from this list that add/subtract to '10' . They are '2' and '8'.

Setting up brackets

( x 2)(x 8)

Since '16' is positive(+), then both signs must be the same. Since '10x' is negative (-) , then both signs are negative.

Hence

(x - 2)(x - 8 ) = 0

When

x - 2 = 0 x = 2

& when

x - 8 = 0

x = 8

So the two answers are ,2, & ,8, .

If x = 2 , then it is a vertical line, parallel with the y -axis.

The gradient is described as 'undefined'.

3y^(4)- y^(2) = 0

Factor

y^(2) [ 3y^(2) - 1] = 0

Hence

y^(2) = 0

It follows that y = 0

And

3y^(2) - 1 = 0

y^(2) = 1/3

y = +/- sqrt( 1/3) = +/-0.577350269.....

Hence y = 0 or y = +/-0.577350269.... to make the eq'n equal to zero.

4x - y = 1

3x + y = 13

Simultaneous Equations.

Add ; This will eliminate 'y'

Hence

7x = 14

x = 2

When x = 2

3(2) + y = 13

6 + y = 13

y = 13 - 6

y = 7

Hence the solution in ( x,y) form is ( 2, 7).

To solve a proportion with a missing variable, you can use cross-multiplication. A proportion means two ratios are equal. For example, if

3

/

4

=

𝑥

/

8

3/4=x/8, multiply across the fractions:

3

×

8

=

4

×

𝑥

3×8=4×x. This gives

24

=

4

𝑥

24=4x. Then divide both sides by 4 to find

𝑥

=

6

x=6. This simple method works for most proportion problems and helps students find the missing value quickly. For step-by-step explanations of problems like this, many students use Sorry Teacher to understand the concept more clearly.

Simultaneous Equations.

4x + 5y = 51

4x - 5y = 21

eliminate 'y' by addition.

Hence

8x = 72

x = 72/8

x = 9

When x = 9 Substitute into eith equation.

4(9) + 5y = 51

36 + 5y = 51

5y = 51 - 36

5y = 15

y = 3

Hence the answer is ( x,y) = ( 9,3).

-38n = 46

Divide both sides by ' -38'

Hence

n = 46/ -38

Cancel dowen by '2'

n = 23/ -19

n = -1.210526316.... ( Note the answer is negative)/.

2x + 2 = x + 5

Subtract '2' from both sides

Hence

2x = x + 3

Subtract 'x' from both sides

x = 3 The answer !!!!!

2x + 8 = 2(x+4)

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