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Flura [38]
3 years ago
11

A line with a slope of 4 and point (2, 1). Find the y-

Mathematics
1 answer:
Nana76 [90]3 years ago
3 0

Answer: y=4x-7

Step-by-step explanation:

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Derivative of tan(2x+3) using first principle
kodGreya [7K]
f(x)=\tan(2x+3)

The derivative is given by the limit

f'(x)=\displaystyle\lim_{h\to0}\frac{f(x+h)-f(x)}h

You have

\displaystyle\lim_{h\to0}\frac{\tan(2(x+h)+3)-\tan(2x+3)}h
\displaystyle\lim_{h\to0}\frac{\tan((2x+3)+2h)-\tan(2x+3)}h

Use the angle sum identity for tangent. I don't remember it off the top of my head, but I do remember the ones for (co)sine.

\tan(a+b)=\dfrac{\sin(a+b)}{\cos(a+b)}=\dfrac{\sin a\cos b+\cos a\sin b}{\cos a\cos b-\sin a\sin b}=\dfrac{\tan a+\tan b}{1-\tan a\tan b}

By this identity, you have

\tan((2x+3)+2h)=\dfrac{\tan(2x+3)+\tan2h}{1-\tan(2x+3)\tan2h}

So in the limit you get

\displaystyle\lim_{h\to0}\frac{\dfrac{\tan(2x+3)+\tan2h}{1-\tan(2x+3)\tan2h}-\tan(2x+3)}h
\displaystyle\lim_{h\to0}\frac{\tan(2x+3)+\tan2h-\tan(2x+3)(1-\tan(2x+3)\tan2h)}{h(1-\tan(2x+3)\tan2h)}
\displaystyle\lim_{h\to0}\frac{\tan2h+\tan^2(2x+3)\tan2h}{h(1-\tan(2x+3)\tan2h)}
\displaystyle\lim_{h\to0}\frac{\tan2h}h\times\lim_{h\to0}\frac{1+\tan^2(2x+3)}{1-\tan(2x+3)\tan2h}
\displaystyle\frac12\lim_{h\to0}\frac1{\cos2h}\times\lim_{h\to0}\frac{\sin2h}{2h}\times\lim_{h\to0}\frac{\sec^2(2x+3)}{1-\tan(2x+3)\tan2h}

The first two limits are both 1, and the single term in the last limit approaches 0 as h\to0, so you're left with

f'(x)=\dfrac12\sec^2(2x+3)

which agrees with the result you get from applying the chain rule.
7 0
3 years ago
Which fractions equal the whole number 3?
zalisa [80]

Answer:

9/3=3

Step-by-step explanation:

3 goes into 9   3 times

3,6,9

6 0
3 years ago
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Maria bought 1,000 pounds of bananas for $220. How much did they cost per pound?
densk [106]

Answer:

i believe the answer is $0.22 per pound.

Step-by-step explanation:

8 0
3 years ago
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2.75 + .003 + .158 =
Alecsey [184]


2.75                             Answer is  2.911      

0.003                Make sure you line up the decimal points like shown in example

0.158

_________

2.911

6 0
3 years ago
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Why is 3 + (−5) equal to −2?
DedPeter [7]
It is not equal to -2 because the five is a negative so when adding two different integers are subtracted and with a negative sign.
5 0
3 years ago
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