Finding generic slope, thet slope of the curve at x=c, the tangent line to the curve

[quasar12]

For each find the a.) generic slope using the lim as h approaches 0 at f(x+h)-f(x)/h, b.) the slope of the curve at x=c, c.) the tangent line to the curve at x=c.

1.) y= 10/x-2, c=7 2.) f(x)= square root of (4x+1), c=2


My friend tried to help me but she used the short cut for it and we're not allowed to use it yet. She gave me her answers and they are: #1 generic- m_tan= -10/(x-2)². slope of curve- m_{tan x=7}= -2/5. tangent line- y= -2/5x + 24/5. #2 generic- m_tan= 2/square root(4x+1). slope of curve- m_{tan x=2}= 2/3. tangent line= y=2/3x+ 5/3.

If you can help me that would be great! thank you!

 

[galactus]

\(\displaystyle f(x)=\frac{10}{x-2}\)

Using the definition of the derivative:

\(\displaystyle \frac{\frac{10}{x+h-2}-\frac{10}{x-2}}{h}\)

\(\displaystyle \frac{10(x-2)-10(x+h-2)}{h(x+h-2)(x-2)}\)

\(\displaystyle \displaystyle \lim_{h\to 0}\frac{-10h}{h(x+h-2)(x-2)}\)

Now, can you finish?. Simplify it down to the derivative, then enter in x=7 to find the slope at that point.

 

[quasar12]

thank you so much! could you also help me start f(x)=square root(4x+1), c=2?

 

[HallsofIvy]

Since galactus has just shown you what you need to do, why not try doing it, then come back if you hit a snag?

f(x)= sqrt(4x+ 2). What is f(2)? What is f(2+ h)? What is (f(2+h)- f(2))/h?

What is the limit as h goes to 0?