Hi,

f(x)= -3x^2 + 5x 

f '(x) = -6x + 5, when x = -1   ⇒    f '(-1) =

y = mx + b (representing the Line tangent to the curve at x = -1)

y = 11x + b    |Using Pt(-1,-8) to solve for b

-8 = 11 + b

3 = b

y = 11x + 3 is the line tangent ro the cureve at (-1,-8)

 

Hi,

Note: if (x-3) is a factor, then the Factor Theorem states:  F(3) = 0

f(x)=x^4+12x^3+6x+2

 ≠ 0,  (x-3) NOT a factor

 

Hi,

h(x) =    ||using the Quotient Rule

h '(x) = 

h '(x) = 

h '(x) = 

 

Hi,

h= -16t^2+vt+c

Question states***initial velocity of 158ft/s,launched from atop a 102-foot cliff

 -16t^2+ 158t + 102 = 0        || ax^2 + bx + c = 0

 

 

 

    x = 10.5 sec

 

Hi,

 

Hi,



A right circular cylinder has height 6  and volume 54pie



r^2 = 9,  r = 3  and  

  

 

Hi,

{-25, ...26}  Arithemetic Sequence, d = 1  and 

,   n = 52





0r

 

Hi,

Curious to know where the  5x + 5h + 1 - (5x + 1) came from.

If You meant:

 

Hi,

On a shelf of 24 books how many combinations of arrangement are possible:  24!

 

Hi,

Yes, parenthesis so important

2^(5/2) - 2^(3/2)

 2^(3/2)[2^(2/2) - 1]

 2^(3/2)[2 - 1]

 2^(3/2)

 

Hi,







(23.086, 14.953, 19.302)

 D = 1,  = 23.086,  = 14.953,  = 19.302 



 
Solved by pluggable solver: Using Cramer's Rule to Solve Systems with 3 variables


  

  

  

  

  

  

  First let . This is the matrix formed by the coefficients of the given system of equations.

  

  

  Take note that the right hand values of the system are , , and  and they are highlighted here: 

  

  

  

  

  These values are important as they will be used to replace the columns of the matrix A.

  

  

  

  

  Now let's calculate the the determinant of the matrix A to get . To save space, I'm not showing the calculations for the determinant. However, if you need help with calculating the determinant of the matrix A, check out this solver.

  

  

  

  Notation note:  denotes the determinant of the matrix A.

  

  

  

  ---------------------------------------------------------

  

  

  

  Now replace the first column of A (that corresponds to the variable 'x') with the values that form the right hand side of the system of equations. We will denote this new matrix  (since we're replacing the 'x' column so to speak).

  

  

  

  

  

  

  Now compute the determinant of  to get . Again, as a space saver, I didn't include the calculations of the determinant. Check out this solver to see how to find this determinant.

  

  

  

  To find the first solution, simply divide the determinant of  by the determinant of  to get: 

  

  

  

  So the first solution is 

  

  

  

  

  ---------------------------------------------------------

  

  

  We'll follow the same basic idea to find the other two solutions. Let's reset by letting  again (this is the coefficient matrix).

  

  

  

  

  Now replace the second column of A (that corresponds to the variable 'y') with the values that form the right hand side of the system of equations. We will denote this new matrix  (since we're replacing the 'y' column in a way).

  

  

  

  

  

  

  Now compute the determinant of  to get .

  

  

  

  To find the second solution, divide the determinant of  by the determinant of  to get: 

  

  

  

  So the second solution is 

  

  

  

  

  ---------------------------------------------------------

  

  

  

  

  

  Let's reset again by letting  which is the coefficient matrix.

  

  

  

  Replace the third column of A (that corresponds to the variable 'z') with the values that form the right hand side of the system of equations. We will denote this new matrix  

  

  

  

  

  

  

  Now compute the determinant of  to get .

  

  

  

  To find the third solution, divide the determinant of  by the determinant of  to get: 

  

  

  

  So the third solution is 

  

  

  

  

  ====================================================================================

  

  Final Answer:

  

  

  

  

  So the three solutions are , , and  giving the ordered triple (23.086, 14.953, 19.302)

  

  

  

  

  Note: there is a lot of work that is hidden in finding the determinants. Take a look at this 3x3 Determinant Solver to see how to get each determinant.

  

  

  

 

Hi,

Note:  Power rule: If f(x) =    ⇒     f '(x) =

Be confident and work carefully, my recommendation.

f(x) =  

 f '(x) =

 f '(x) = 6x^(1/2)

 

Hi,

y = x^2 -2x -3       | y = ax^2 + bx + c  and 

y =   |completing the Square to find Vertex form

y = (x-1)^2 - 4 ,    

, the vertex form of a parabola,  where(h,k) is the vertex

The pattern You are seeing is the result of the (x-1)^2 pattern

x   (x-1)^2	y

1	0	-4

2	1	-3

3	4	0

4	9	5

5	16	12

 

Hi,

, the vertex form of a Parabola opening up(a>0) or down(a<0), 

where(h,k) is the vertex  and  x = h  is the Line of Symmetry

Why does the x^2 coefficient of the equation of a parabola determine the direction it opens...

As the value of the 'squared' term (which is always positve) ->infinity,

that coefficient will determine which direction the parabloa heads (to +infinity or -infinity)



Same Idea applies to a parabola open right or left

,the vertex form of a Parabola opening right(a>0) or left(a<0), 

 where(h,k) is the vertex and  y = k  is the Line of Symmetry

 

Hi,

in how many ways can 3 boys and 2 girls sit in a row? 

Thinking of the boys and girs being 1 unit: 2! 

Girls themselves can be arranged in 2 ways

Boys themselves can be arranged in 3! ways

in how many ways can 3 boys and 2 girls sit in a row? 

 

 

Hi,

What is the value of the slope of the tangent to the curve at x = +2

f (x) = x ^ 3 - x ^ 2 - 2x      | Taking the 1st Derivative    

f '(x) =  

f '(2) =  , the slope-value 

y = mx + b

     |Using (x,y) coordinate (2,0) to solve for b, 

 0 = 12 + b

-12 = b

EQ of the tangent line at x = 2 is y = 6x-12

 

Hi,

   |Multiplying thru by 2 so as all denominators = 1

    y + 12  = 4y + 42

       -30 = 3y

       -10 = y

and...

 

 

Hi,

1 1/3 times a number is equal to 6

 

 

  

     a = 18/4 = 9/2  or 

 

Hi,

log 12x^2 - log 3x = 3

0r

log 12x^2/3x = 3

  10^3 = 4x

 1000/4 = x

 

Hi,



 

0r  

    8m by 8m

   10m by 6m

etc

   

 

Hi,

  0r 

starting at adding 2,  lands one on 

 

Hi,

2/x + 1/y = 0

 5/x + 2/y = -5 

'u' = 1/x and 'v' = 1/y ****



2u + v = 0   |multiplying thru by -2 to  eliminate the v variable

5u + 2v = -5 

  u = -5   and v = 10 

 x = 1/-5   and v = 1/10

and...

 

 

Hi,

(1, 9500)

(0, 15000)   m = (9500-15000)/1 = -5500

Value = -5500x + 15000 , where x is the number of years from 2010.

 

Hi,

 (x^3 - 13x^2 - 41x + 4) / (x - 7)

may use Synthetic Divsion for this ex.

7	1	-13	-41	4

 	 	7	-42	-581

 	1	-6	-83	-577

Yields x^2 - 6x - 83    R = -577

 

Hi,

C=0.4x^2+16x+10  x > 0

C = .4x + 16 + 10/x 

C' = .4  - 10/x^2

   .4 - 10/x^2 = 0

       x^2 = 10/.4 = 25 

         x = 5, units that should be produced to minimize the average cost per unit

 

Hi,

1/4x+1/3y= 5    || |multiplying thru by 3 to  eliminate the y variable 

 x - y= 6             





   x = 

    x = 12    and  y = 6

and

 

 

 

Hi,

The 'clue' to the problem is the numbers must end with something that

results in a zero in the hundredths place when multiplied: 

x + y = 2.7



 x^2 - 2.7x + 1.8 = 0   |

 

 x =  1.5  0r  x = 1.2

 

 

Hi,





 x= 2 ⇒ (15.32 -17.45)

 x = 0 ⇒  -11.12

  (15.32 -17.45) + 11.12 = 8.99

 

 

Hi,

g(x)= x^2 + x -7/3 = (x+1/2)^2 - 1/4 - 7/3 =(x+1/2)^2 - 31/12 

domain: all real numbers.

range: g(x) ≥  - 31/12

 

Hi,

4x^2-8x+3=0

(2x-1)(2x-3) = 0  x=intercepts are 1/2, 3/2

vertex form of a Parabola opening up(a>0) or down(a<0),  

where(h,k) is the vertex  and  x = h  is the Line of Symmetry

4x^2-8x+3= 4(x-1)^2 -4 + 3 = 4(x-1)^2 -1  

V(1,-1), parabola opens Upward, x= 1 the line of symmetry

 

Hi,

solve for x and y by using elimination.

 3x+5y=-1      |Multiplying 1st EQ by 4, 2nd by 3 and subtracting to eliminate  x

 4x+7y=4     

  -y = -16

   y = 16   and x = -27

and...