Find the slope and equation of the tangent line to the graph of the function at the given value of x.f(x)=x^4-20x^2+64;x=-1

Answers

Answer 1

Answer: The slope is the differential of the function.

Recall, if y = x^n, (dy/dx) = nx^(n-1).

y=x^4-20x^2+64; x = -1. To differentiate this, we do it for each term.

(dy/dx) = (4)(x^(4 -1)) - (2)(20x^(2-1) + 0*64x^(0-1) (Note 64 = 64x^0, x^0 =1)
   = (4)x^(3) - 40x^(1) + 0
      = 4x^3 - 40x^1.

(dy/dx) = 4x^3 - 40x. Note at x = -1.

(dy/dx), x = -1, =     4(-1)^3 - 40(-1)
   = -4 + 40 = 40 - 4 = 36.

Slope at x = -1 is 36.
Cheers.

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Describe this pattern. Then see if you can think of another Pythagorean triple that doesn't follow the pattern you just described and that can't be generated using the identity (x2 − 1)2 + (2x)2 = (x2 + 1)2. Explain your findings. x-value Pythagorean Triple 7 (14,48,50) 8 (16,63,65) 9 (18,80,82) 10 (20,99,101)

Answers

Answer:

See explanation

Step-by-step explanation:

A Pythagorean triple is a set of 3 positive integer numbers, a, b, and c which satisfies the Pythagorean theorem: $c^2=a^2+b^2$

Given the identity: $(x^2 - 1)^2 + (2x)^2 = (x^2 + 1)^2.$

We verify that the numbers given table are Pythagorean triples.

$14^2+48^2=2500=50^2\n16^2+63^2=4225=65^2\n18^2+80^2=6724=82^2\n20^2+99^2=10201=101^2$

Next, we examine the pattern:

$\left|\begin{array}{c|c|c|c|c}x-value&x^2-1&2x&x^2 + 1&$Triple\n--&--&--&--&----\n7&48&14&50&(14,48,50)\n8&63&16&65&(16,63,65)\n9&80&18&82&(18,80,82)\n10&99&20&101&(20,99,101)\end{array}\right|$

From the pattern, our first number is 2x, the second number is x² - 1, and the third number(hypotenuse) is x² + 1

Next, we show that (x² - 1)² + (2x)² = (x² + 1)² is an identity

$LHS: (x^2 - 1)^2 + (2x)^2 \n= (x^2-1)(x^2-1)+2x^2\n=x^4-2x^2+1+4x^2\n=x^4+2x^2+1$

$RHS: (x^2+1)^2\n= (x^2+1) (x^2+1)\n=x^4+2x^2+1$

Since Left Hand Side=Right hand Side. For all x, the equation is always true.

Therefore, the pattern is true for any Pythagorean triple.

How do you simplify the cube root of 32?

Answers

I hope this helps you

Elliott has 4 columns of marbles. He has 3 marbles in each column. How many marbles does he have in all? *

Answers

Answer:

3*4=12

Step-by-step explanation:

the answer is 12

Final answer:

To find out how many marbles Elliott has in total, we multiply the number of columns by the number of marbles in each column, resulting in 12 marbles in total.

Explanation:

To find out how many marbles Elliott has in total, we can multiply the number of columns by the number of marbles in each column. In this case, Elliott has 4 columns with 3 marbles in each column. So, we can calculate the total number of marbles by multiplying 4 and 3, which equals 12. Therefore, Elliott has 12 marbles in total.

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Х+5y = 3
3х - 2y = -8

Answers

X equals -2 and y equals 1

Solve for X: x+5y=3

Subtract 5y from both sides: x=3−5y

Substitute
x=3−5y into 3x−2y=−8

Start with the original equation: 3x−2y=−8

Let x=3−5y: 3(3−5y)−2y=−8

Simplify: 9−17y=−8

Solve for y in 9−17y=−8: y=1

Substitute y=1 into , x=3−5y : x=-2

X=-2
Y=1

Use elimination to solve this system of equations. 2r + 3s= 9 and 3r + 2s= 12

Answers

2r + 3s= 9 <=> -6r -9s = -27;
3r + 2s= 12 <=> 6r + 4s = 24;
=> 0*r -5*s = -3 => s = 3/5;

2*r + 3*(3/5) = 9<=> 10*r + 9 = 45 <=> 10*r = 36 => r = 18/5;

Answer:

(18/5,3/5)

Step-by-step explanation:

Abcd is a rectangle. Find the length of each diagonal. .AC= 3y/5 BD=3y-4