Question

The point P(9 , 6 ) lies on the curve y = \sqrt{ x } + 3. Let Q be the point (x, \sqrt{ x }+ 3 ).

a.) Find the slope of the secant line PQ for the following values of x(Answers here should be correct to at least 6 places after the decimal point.)

If x= 9.1, the slope of PQ is:
If x= 9.01, the slope of PQ is:
If x= 8.9, the slope of PQ is:
If x= 8.99, the slope of PQ is:

b.) Based on the above results, estimate the slope of the tangent line to the curve at P(9 , 6 ).

Answer 1

If x= 9.1, then what is the value of y?

Answer 2

Of the original function or the derivative?

Answer 3

original function

Answer 4

y = \sqrt{ x } + 3 at x=9.1 --> 6.016620626

Answer 5

so we know the point (9.1, 6.016620626) lies on the function curve

Answer 6

P = (9,6)
Q = (9.1, 6.016620626)

slope of PQ = ??

Answer 7

that's what I don't get, I do not know what to do...

Answer 8

you would use the slope formula

\[\LARGE m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}\]

Answer 9

it doesn't work because both x are 9 (we cannot divide by 0)

Answer 10

x1 = 9
x2 = 9.1

Answer 11

so x2 - x1 = 9.1 - 9 = 0.1

Answer 12

oh i missed that one, the answer is 0.16620626 ... would that be the answer for the first one?

Answer 13

I entered it and it says correct!! thank you! what about b) ?

Answer 14

y1 = 6
y2 = 6.016620626

y2 - y1 = 6.016620626 - 6 = 0.016620626

divide that by 0.1 and you get 0.016620626/0.1 = 0.16620626

so it looks good

Answer 15

repeat the same steps as a) but now use x = 9.01 instead of 9.1

Answer 16

I meant the second part of the question

Answer 17

tell me what slopes you get for part a)

Answer 18

for every one of them of just the first one?

Answer 19

each one

Answer 20

0.16620626 -0.1666203961-0.1671322196-0.1667129887

Answer 21

We just do the average?

Answer 22

it's probably not 100% clear but what do you notice that is common to all of these slopes?

Answer 23

I tried the average of them and it is correct! Thank you so much, I appreciate it greatly! Could you help me with another quick question?

Answer 24

sure

Answer 25

Let f(x) = 3x^{4}-4.

(a) Use the limit process to find the slope of the line tangent to the graph of f at x = 3.

Slope at x = 3: (I got 324 but i dont know if it is correct)

(b) Find an equation of the line tangent to the graph of f at x = 3.

Tangent line: y =

Answer 26

Do you know how to help me on this one?