1. If = 1, = −2 = −1 then find the value of the following

Derivative, by first principle<br><img src="https://tex.z-dn.net/?f=%20%5Ctan%28%20%5Csqrt%7Bx%20%7D%20%29%20" id="TexFormula1"

vampirchik [111]

$\displaystyle\lim_{h\to0}\frac{\tan\sqrt{x+h}-\tan x}h$

Employ a standard trick used in proving the chain rule:

$\dfrac{\tan\sqrt{x+h}-\tan x}{\sqrt{x+h}-\sqrt x}\cdot\dfrac{\sqrt{x+h}-\sqrt x}h$

The limit of a product is the product of limits, i.e. we can write

$\displaystyle\left(\lim_{h\to0}\frac{\tan\sqrt{x+h}-\tan x}{\sqrt{x+h}-\sqrt x}\right)\cdot\left(\lim_{h\to0}\frac{\sqrt{x+h}-\sqrt x}h\right)$

The rightmost limit is an exercise in differentiating $\sqrt x$ using the definition, which you probably already know is $\dfrac1{2\sqrt x}$ .

For the leftmost limit, we make a substitution $y=\sqrt x$ . Now, if we make a slight change to $x$ by adding a small number $h$ , this propagates a similar small change in $y$ that we'll call $h'$ , so that we can set $y+h'=\sqrt{x+h}$ . Then as $h\to0$ , we see that it's also the case that $h'\to0$ (since we fix $y=\sqrt x$ ). So we can write the remaining limit as

$\displaystyle\lim_{h\to0}\frac{\tan\sqrt{x+h}-\tan\sqrt x}{\sqrt{x+h}-\sqrt x}=\lim_{h'\to0}\frac{\tan(y+h')-\tan y}{y+h'-y}=\lim_{h'\to0}\frac{\tan(y+h')-\tan y}{h'}$

which in turn is the derivative of $\tan y$ , another limit you probably already know how to compute. We'd end up with $\sec^2y$ , or $\sec^2\sqrt x$ .

So we find that

$\dfrac{\mathrm d\tan\sqrt x}{\mathrm dx}=\dfrac{\sec^2\sqrt x}{2\sqrt x}$

7 0

3 years ago

A pair of pants with a marked price of $35.00 has a discount of 20%. How much is the discount? Can someone help me learn to do p

andreyandreev [35.5K]

When you see a "%" you need to ask " a percent of what"?

Here, it's a percent of the original prize, so the discount is 20% of the original prize, that is 20% of 35.

20% is also 20/100 or 1/5, so we can calculate this like this:

$\frac{1}{5} *35=7$

so the discount is 7 dollars.

6 0

3 years ago

Find the value of $B - A$ if the graph of $Ax + By = 3$ passes through the point $(-7,2),$ and is parallel to the graph of $x +

Salsk061 [2.6K]

Answer:

-6

Step-by-step explanation:

We know that since Ax + By = 3 passes through (-7, 2), then if we plug -7 in for x and 2 in for y, the equation is satisfied. So, let's do that:

Ax + By = 3

A * (-7) + B * 2 = 3

-7A + 2B = 3

We also know that this line is parallel to x + 3y = -5, which means their slopes are the same. Let's solve for y in the second equation:

x + 3y = -5

3y = -x - 5

y = (-1/3)x - (5/3)

So, the slope of this line is -1/3, which means the slope of Ax + By = 3 is also -1/3. Let's solve for y in the first equation:

Ax + By = 3

By = -Ax + 3

y = (-A/B)x + 3/B

This means that -A/B = -1/3. So, we have a relationship between A and B:

-A/B = -1/3

A/B = 1/3

B = 3A

Plug 3A in for B into the equation we had where -7A + 2B = 3:

-7A + 2B = 3

-7A + 2 * 3A = 3

-7A + 6A = 3

-A = 3

A = -3

Use this to solve for B:

B = 3A

B = 3 * (-3) = -9

So, B = -9 and A = -3. Then B - A is:

B - A = -9 - (-3) = -9 + 3 = -6

The answer is -6.

<em>~ an aesthetics lover</em>

7 0

3 years ago

Read 2 more answers

What is the result of isolating y2 in the equation below?

lyudmila [28]

STEP-BY-STEP SOLUTION:

( x + 4 )^2 + y^2 = 22

x^2 + 2 × x × 4 + 4^2 + y^2 = 22

x^2 + 8x + 16 + y^2 = 22

y^2 = 22 - x^2 - 8x - 16

y^2 = - x^2 - 8x + 6

FINAL ANSWER:

Therefore, the answer is:

D. y^2 = - x^2 - 8x + 6

Please mark as brainliest if you found this helpful! :)
Thank you <3

6 0

3 years ago

ericas mother saved her quarters and dimes .she counted 48 coins with a value of $7.50.how many quarters and dimes were there？

katrin2010 [14]

28 Quarters and 5 dimes

8 0

3 years ago

Read 2 more answers