anonymous
  • anonymous
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 ).
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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jim_thompson5910
  • jim_thompson5910
If x= 9.1, then what is the value of y?
anonymous
  • anonymous
Of the original function or the derivative?
jim_thompson5910
  • jim_thompson5910
original function

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anonymous
  • anonymous
y = \sqrt{ x } + 3 at x=9.1 --> 6.016620626
jim_thompson5910
  • jim_thompson5910
so we know the point (9.1, 6.016620626) lies on the function curve
jim_thompson5910
  • jim_thompson5910
P = (9,6) Q = (9.1, 6.016620626) slope of PQ = ??
anonymous
  • anonymous
that's what I don't get, I do not know what to do...
jim_thompson5910
  • jim_thompson5910
you would use the slope formula \[\LARGE m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}\]
anonymous
  • anonymous
it doesn't work because both x are 9 (we cannot divide by 0)
jim_thompson5910
  • jim_thompson5910
x1 = 9 x2 = 9.1
jim_thompson5910
  • jim_thompson5910
so x2 - x1 = 9.1 - 9 = 0.1
anonymous
  • anonymous
oh i missed that one, the answer is 0.16620626 ... would that be the answer for the first one?
anonymous
  • anonymous
I entered it and it says correct!! thank you! what about b) ?
jim_thompson5910
  • jim_thompson5910
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
jim_thompson5910
  • jim_thompson5910
repeat the same steps as a) but now use x = 9.01 instead of 9.1
anonymous
  • anonymous
I meant the second part of the question
jim_thompson5910
  • jim_thompson5910
tell me what slopes you get for part a)
anonymous
  • anonymous
for every one of them of just the first one?
jim_thompson5910
  • jim_thompson5910
each one
anonymous
  • anonymous
0.16620626 -0.1666203961-0.1671322196-0.1667129887
anonymous
  • anonymous
We just do the average?
jim_thompson5910
  • jim_thompson5910
it's probably not 100% clear but what do you notice that is common to all of these slopes?
anonymous
  • anonymous
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?
jim_thompson5910
  • jim_thompson5910
sure
anonymous
  • anonymous
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 =
anonymous
  • anonymous
Do you know how to help me on this one?
triciaal
  • triciaal
|dw:1443842056603:dw|
jim_thompson5910
  • jim_thompson5910
keep following @triciaal steps to find the value of f(3) if x = 3, then y = ??
anonymous
  • anonymous
y=239
jim_thompson5910
  • jim_thompson5910
so we know (3,239) lies on the curve
anonymous
  • anonymous
and the slope of 324 was correct?
jim_thompson5910
  • jim_thompson5910
yes
jim_thompson5910
  • jim_thompson5910
so you know this line has a slope of 324 and goes through (3,239)
anonymous
  • anonymous
so y-y1=m(x-x1) ?
jim_thompson5910
  • jim_thompson5910
yes
jim_thompson5910
  • jim_thompson5910
m = 324 (x1,y1) = (3,239)
anonymous
  • anonymous
How do you rearrange it so that the equation looks like y= mx+b ??
jim_thompson5910
  • jim_thompson5910
you just need to solve for y and simplify
anonymous
  • anonymous
It worked! I just made a sign mistake at the end.... @jim_thompson5910 I appreciate your help greatly! you saved my grade! @triciaal Thank you also for helping me I appreciate it
jim_thompson5910
  • jim_thompson5910
you're welcome

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