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Harman [31]
3 years ago
15

Identify each expression that represents the slope of a tangent to the curve y=1/x+1 at any point (x,y) .

Mathematics
1 answer:
tankabanditka [31]3 years ago
6 0

Answer:

Slope of a tangent to the curve = f'(x) = \frac{-1 }{(x+1)^{2} }

Step-by-step explanation:

Given - y = 1/x+1

To find - Identify each expression that represents the slope of a tangent to the curve y=1/x+1 at any point (x,y) .

Proof -

We know that,

Slope of tangent line = f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

We have,

f(x) = y = \frac{1}{x+1}

So,

f(x+h) = \frac{1}{x+h+1}

Now,

Slope = f'(x)

And

f'(x) =  \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}  \\= \lim_{h \to 0} \frac{\frac{1}{x+h+1}  - \frac{1}{x+1} }{h}\\= \lim_{h \to 0} \frac{x+1 - (x+h+1) }{h(x+1)(x+h+1)}\\= \lim_{h \to 0} \frac{x +1 - x-h-1 }{h(x+1)(x+h+1)}\\= \lim_{h \to 0} \frac{-h }{h(x+1)(x+h+1)}\\= \lim_{h \to 0} \frac{-1 }{(x+1)(x+h+1)}\\=  \frac{-1 }{(x+1)(x+0+1)}\\=  \frac{-1 }{(x+1)(x+1)}\\=  \frac{-1 }{(x+1)^{2} }

∴ we get

Slope of a tangent to the curve = f'(x) = \frac{-1 }{(x+1)^{2} }

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Which of the sets of ordered pairs represents a function? (1 point)
Lyrx [107]

Answer:

Only B

Step-by-step explanation:

A has 2 pairs with the x as the number 2, so they are not functions.

B has all independent pairs with their own numbers

So only B is the correct answer.

8 0
2 years ago
75(1+0.2/12)^10*12 what is the answer
Alchen [17]

Answer:

545.119

Step-by-step explanation:

75 * (1 + .2/12)^10 * 12

75 * (1.016667)^10*12

Now things get a little tricky. What does 10*12 mean? I'm taking it to mean 120

75 * (1.016667)^120

75 * 7.268

545.119

8 0
3 years ago
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6. Blake went to the grocery store at 6:30PM. He returned to his house around 9:14PM after running
babunello [35]

Answer:

The answer is 2:44 minutes. I had this question and got it right. You have to subtract 9:14 - 6:30 = 2:44. Have a great day.

Step-by-step explanation:

4 0
3 years ago
Find the rate of change of f(x, y, z) = xyz in the direction normal to the surface yx2 + xy2 + yz2 = 3 at (1, 1, 2).
Paladinen [302]

Answer:

Rate of change = 18/√74

Step-by-step explanation:

The rate of change of f at x_o in the direction of the unit vector v is given by: ∇f(x_o) × v

Now, for us to get the direction of the unit vector, since we are told that f(x, y, z) = xyz in the direction normal to the surface yx² + xy² + yz² = 3, we will use; g(x, y, z) = yx² + xy² + yz² = 3 and k =3 to give;

∇g(x,y,z) = (2xy + y², (x² + 2xy + z²), 2yz)

So;

∇g(1, 1, 2) = (2(1 × 1) + 1²) , (1² + 2(1 × 1) + 2²), 2(1 × 2))

This gives;

∇g(1, 1, 2) = (3, 7, 4)

Now,

||∇g(1, 1, 2)|| = √(3² + 7² + 4²)

||∇g(1, 1, 2)|| = √74

Normal vector would be given by the formula;

v = [∇g(1, 1, 2)]/[||∇g(1, 1, 2)||]

Thus;

v = (3, 7, 4)/√74

Let's not forget that our ∇f(1, 1, 2) = (1, 1, 2)

Thus, rate of change is given by;

∇f(1, 1, 2) × v = (1, 1, 2) × (3, 7, 4)/√74

This gives;

Rate of change = [(1 × 3) + (1 × 7) + (2 × 4)]/√74 = 18/√74

7 0
3 years ago
Graded Assignment Unit Test, Part 2: More Math Models Answer the questions. When you have finished, submit this assignment to yo
Black_prince [1.1K]

Answer:

Y(n) = 400(2.5)^n

Y = 3,814,697 rice weevils

16.1 weeks

Step-by-step explanation:

To predict the population of rice weevil for any week after the start of the study ; wwe use an exponential function for this purpose :

y(n) = A(1 + r)^n

Where;

Y(n) = population at week n

r = growth rate ; n = number of weeks

A = initial population at the start of the study

r = 150% = 1.5

Hence, we have the equation :

Y(n) = 400(1 + 1.5)^n

Y(n) = 400(2.5)^n

B.)

Population after 10 weeks

n = 10

Y(10) = 400(2.5)^10

Y = 400 * 9536.7431640625

Y = 3814697.265625

Y = 3,814,697 rice weevils

C)

When population will reach 1,000,000,000

We have :

1,000,000,000 = 400(2.5)^n

1000000000 / 400 = 2.5^n

2500000 = 2.5^n

Take the log of both sides :

log(2500000) = nlog(2.5)

6.3979400 = n * 0.3979400

n = 6.3979400 / 0.3979400

n = 16.077649

n = 16.1 weeks approximately

5 0
3 years ago
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