Simplify: 6t + t Answer: __

Let f(x) = 1/x^2 (a) Use the definition of the derivatve to find f'(x). (b) Find the equation of the tangent line at x=2

Verdich [7]

Answer:

(a) $f'(x)=-\frac{2}{x^3}$

(b) $y=-0.25x+0.75$

Step-by-step explanation:

The given function is

$f(x)=\frac{1}{x^2}$ .... (1)

According to the first principle of the derivative,

$f'(x)=lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}$

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

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

$f'(x)=lim_{h\rightarrow 0}\frac{x^2-x^2-2xh-h^2}{hx^2(x+h)^2}$

$f'(x)=lim_{h\rightarrow 0}\frac{-2xh-h^2}{hx^2(x+h)^2}$

$f'(x)=lim_{h\rightarrow 0}\frac{-h(2x+h)}{hx^2(x+h)^2}$

Cancel out common factors.

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

By applying limit, we get

$f'(x)=\frac{-(2x+0)}{x^2(x+0)^2}$

$f'(x)=\frac{-2x)}{x^4}$

$f'(x)=\frac{-2)}{x^3}$ .... (2)

Therefore $f'(x)=-\frac{2}{x^3}$ .

(b)

Put x=2, to find the y-coordinate of point of tangency.

$f(x)=\frac{1}{2^2}=\frac{1}{4}=0.25$

The coordinates of point of tangency are (2,0.25).

The slope of tangent at x=2 is

$m=(\frac{dy}{dx})_{x=2}=f'(x)_{x=2}$

Substitute x=2 in equation 2.

$f'(2)=\frac{-2}{(2)^3}=\frac{-2}{8}=\frac{-1}{4}=-0.25$

The slope of the tangent line at x=2 is -0.25.

The slope of tangent is -0.25 and the tangent passes through the point (2,0.25).

Using point slope form the equation of tangent is

$y-y_1=m(x-x_1)$

$y-0.25=-0.25(x-2)$

$y-0.25=-0.25x+0.5$

$y=-0.25x+0.5+0.25$

$y=-0.25x+0.75$

Therefore the equation of the tangent line at x=2 is y=-0.25x+0.75.

5 0

3 years ago

Explain in 1-2 sentences how you identify

andrezito [222]

Answer:

You can tell because, as x increases/decreases, y increases/decreases at the same rate EACH TIME. This way, no matter how many times the proportions increase OR decrease, the ratio between x and y always stays the same.

8 0

2 years ago

Give the function y=-5x+12, what is the output (y) when the input, x=-2?

artcher [175]

Answer:

y=22

Step-by-step explanation:

y=-5(-2)+12

y=10+12

y=22

6 0

2 years ago

Algebra math word problems

irga5000 [103]

26. 4 people went to the aquarium

27. $1.50 per game (subtraction property and division property)

7 0

3 years ago

PLZZZ CHECK ME Is this correct, if not please help me.

Gwar [14]

Hello! For this question, you solve using order of operations, which you probably already have remembered as PEMDAS. Anytime a number is raised to the 0 power, that is equivalent to 1. 0.5^0 is 1. 250 * 1 is 250. x = 0 gives us the answer of 250. Now, let's solve for the number by the second power. 0.5² is 0.25. 250 * 0.25 is 62.5. x² gives us 62.5. Now, let's subtract the two numbers together. 250 - 62.5 is 187.5. Because the number is decreasing, the slope is negative, so the answer is -187.5. Your answer is correct.

5 0

3 years ago

Simplify: 6t + tAnswer: ___t​

Simplify: 6t + tAnswer: ___t