## anonymous 3 years ago helpppp: find the critical points: s(t)= (t-1)^4 (t+5)^3

1. anonymous

@mathcalculus , what have you tried?

2. anonymous

yes, i used the product rule: $\frac{ d }{ dx } (t-1)^{4}* (t+5)^{3}= (t-1)^{4} * 3(t+5)^{2} + (t+5)^{3} * 4(t-1)^{3}$

3. anonymous

then from there, i'm stuck.

4. anonymous

ok, so far so good. Now, factor out the greatest common factor of both terms. For example, they both have a factor of (t-1)^3, but that's not all they have in common. What else?

5. anonymous

?

6. anonymous

Don't they both have a factor of t+5 to a power?

7. anonymous

oh yes, but are from opposite sides..

8. anonymous

doesn't matter the order. AB+CA=A(B+C)

9. anonymous

can you explain to me. i dont really see where we are going with factoring

10. anonymous

The purpose in factoring is to simplify the derivative into a product of three factors. One of the factors will be t-1 to a power, a second factor will be t+5 to a power, and then you will be left with some factor containing a t as well.

11. anonymous

And then once you have your derivative in factored form, you can set each factor equal to zero to find your critical points.

12. anonymous

can you show me

13. anonymous

i understand what youre saying but not how sure how to do that.

14. anonymous

Ok, lets look at the t-1 term in each expression.

15. anonymous

ok

16. anonymous

What is the largest exponent they share?

17. anonymous

3

18. anonymous

now i know that i write (t-1)^3

19. anonymous

but what happens to the 4?

20. anonymous

Exactly, so, we may factor out a $\left( t-1 \right)^{3}$ from both expressions. Hang on, we will come to that.

21. anonymous

k

22. anonymous

Let's take a looke at the first expression in its entirety. If all you are removing is the $\left( t-1 \right)^{3}$ You would be left with $4\left( t+5 \right)^{3}$

23. anonymous

Do you see that?

24. anonymous

no if i was to see the common factor.. i would get (t-1)^3+ (t+5)^2.... then what do i do with the numbers that are left...?

25. anonymous

ok,, there is your mistake. It isn't a sum. It's a product.

26. anonymous

ok so i dont see how you got 4(t+5)^3

27. anonymous

Hang on, I was only talking about factoring out the t-1 from the first expression,, not from the whole thing.

28. anonymous

well you showed me that

29. anonymous

Hang on my pc is giving me fits.

30. anonymous

kk

31. anonymous

|dw:1363286087022:dw|

32. anonymous

why 4(t+5)?????

33. anonymous

On the outside is what they both have in common, on ther inside is what is left after you have factored out that common factor.

34. anonymous

i understand this part... (t-1)^3* (t+5)^2

35. anonymous

Rememmber when I was saying if all we factored out was the (t-1)^2 You would have the 4(t+5)^3 left.

36. anonymous

but then i dont understand why it is 4(t+5) ... i did this..4(t-1) + 3(t+5)

37. anonymous

Well, now we include a (t+5)^2 as well, so what is left from the 4(t+5)^3? Just the 4(t+5).

38. anonymous

okkk

39. anonymous

i understand...

40. anonymous

im left with 7t+17

41. anonymous

t= -17/7

42. anonymous

Perfect. So, now that each expression of the derivative is writtern in factored form, we set each of them equal to zero to find your critical points.

43. anonymous

That is one of your critical points. There are two others.

44. anonymous

ok so (t-1)^3 = 0

45. anonymous

Good.

46. anonymous

x= 0?

47. anonymous

Not quite.

48. anonymous

If (t-1)^3=0 then t=1

49. anonymous

Now you have one more to do.

50. anonymous

1? how?

51. anonymous

Tinme for lunch, I will come back shortly to see if you have finished it.

52. anonymous

the exponent is in the way.

53. anonymous

Well, raise each side to the 1/3 power. That will get rid of your exponent.

54. anonymous

brb

55. anonymous

kkk but how do we find the y

56. anonymous

Just go back in and substitute the critical values.

57. anonymous

For t=1 and t=-5, you will get y=0. But for t=-17/7, you won't get y=0.

58. anonymous

i did....

59. anonymous

(1-1)^4 (1+5)^3

60. anonymous

=0

61. anonymous

and -17/7 i got a huge number.. doesn;t make sense.. :(

62. anonymous

i got -432/823543

63. anonymous

Well, you will need to raise some fractions with a denominator of 7 to the 4th and 3rd powers, so yeah, it will get rather ugly!

64. anonymous

??

65. anonymous

i see that but it keeps saying the critical points are wrong.

66. anonymous

=(

67. anonymous

ok, I need to see the original problem. And what keeps saying your critical points are wrong?

68. anonymous

Are you using an online program?

69. anonymous

webwork.

70. anonymous

yes

71. anonymous

hang on.

72. anonymous

73. anonymous

i attached it.

74. anonymous

Your value for t=-17/7 is incorrect

75. anonymous

why?

76. anonymous

You should get a large numerator divided by a denominator of 7^7.

77. anonymous

so i dont know where i went wrong...

78. anonymous

The numerator should be (-24)^4(18)^3

79. anonymous

we did this step by step... now t is wrong?? really?

80. anonymous

No, t is not wrong, but your evaluation of the function for t=-17/7 is incorrect.

81. anonymous

so now how do i solve that?

82. anonymous

obviously i know its wrong.

83. anonymous

ive tried this problem out 3294810347 times.

84. anonymous

now what do i do?

85. anonymous

Should have (-17/7-1)^4(-17/7+5)^3

86. anonymous

huhhh???

87. anonymous

$-\frac{ 17 }{ 7 }-1=-\frac{ 24 }{ 7 }$ and $-\frac{ 17 }{ 7 }+5=\frac{ 18 }{ 7 }$

88. anonymous

wow this is insane. ive spent an hour to 2 with this problem.

89. anonymous

why?? why did we ignore the exponents???

90. anonymous

Now raise each of those terms to the appropriate powers.

91. anonymous

sn;t it suppose to be part of it?? the original problem we were given??

92. anonymous

i did

93. anonymous

we didn't I was just showing you step by step.

94. anonymous

thats how i got the hugeeeeeeee number

95. anonymous

and so did i. i had it on paper.

96. anonymous

what i dont understand is that youre basically saying raise those to the power.. in other words, im redoing this whole crap again.

97. anonymous

98. anonymous

99. anonymous

No, you are finding the y-value that goes with the value you ccame out for t.

100. anonymous

so there are 4 crtical points now?

101. anonymous

or 3??

102. anonymous

youre saying raise hem to the power and multiple right???

103. anonymous

do you know rasing them to the power is going to GIVE ME A HUGEEEEEE NUMBER??

104. anonymous

do you not see that was what i had in the first place??

105. anonymous

There are only the three values of t we came up with, but you want them as ordered pairs, yo

106. anonymous

ok good3 values done.

107. anonymous

obviously i want them in order pairs.

108. anonymous

i did EXACTLY what you just did with the -17/7 and got those numbers.

109. anonymous

I see that, but your answer for t=-17/7 showed a smaller numerator than denominator and it shouldd be the other way around.

110. anonymous

then show me.

111. anonymous

whats the y value for -17/7

112. anonymous

How did you do that? Each numerator is larger than each denominator and they both have total of 7th powers.

113. anonymous

114. anonymous

i need to see if it's correct

115. anonymous

i cant keep wasting 2-3 hours on one problem.

116. anonymous

@calcmat, are you stuck?

117. anonymous

$\left(- \frac{ 24 }{ 7 } \right)^{4}\times \left( \frac{ 18 }{ 7} \right)$

118. anonymous

yeah

119. anonymous

now 18/7 needs to be raised to the 3rd power.

120. anonymous

ACCORDING TO THE ORIGINAL PROBLEM

121. anonymous

Should gha have a 3 on the second expression for the power.

122. anonymous

and i just proved my point. you are taking em to the wrong direction AGAIN.

123. anonymous

Correct.

124. anonymous

125. anonymous

126. anonymous

I WANT TO SEE IF WHAT YOU'RE DOING IS ACTUALLY CORRECT

127. anonymous

IVE SPENT 2-3 HOURS WITH YOU IN THIS PROBLEM.

128. anonymous

Hey, don't get mad at me, I am the one spending MY TIME helping YOU.

129. anonymous

YOURE STILL NOT ANSWERING MY QUESTION.

130. anonymous

131. anonymous

I DID. STILL A HUGE NUMBER

132. anonymous

I got 2349.5 rounded to 1 decimal place.

133. anonymous

WRONG BUDDY.

134. dan815

(-1+t)^3 (5+t)^2 (17+7 t) your 0's = 1,-5,-17/7

135. anonymous

nvm. i''ll solve it

136. anonymous

thanks everyone.

137. anonymous

Those are the ones I got @dan815

138. anonymous

that part is correct.

139. anonymous

and @calmat01 don't round. thats wrong.

140. anonymous

keep the numbers just the way they are. p.s found the answer. thanks.

141. anonymous

Ok, so my answer wasn't completely off?

142. dan815

mathcalculus can you prove the divergence theorem for me?

143. anonymous

So you are telling me that you should have left the answer as 1934917632/823543? That's insane @mathcalculus

144. dan815

what are all these big numbers?

145. anonymous

yes.

146. anonymous

lol insane.

147. anonymous

your explanation was confusing. but the numbers are correct.

148. anonymous

Gee, I spent all that time being patient with you, and all I get is my explanation was confusing?

149. anonymous

lol truth to be told...truth be told.

150. anonymous

I see. Well, if truth be told, the reason it was confusing was because you forgot how to factor. But I am not bitter. Take care and good luck with the rest of your assignment.

151. anonymous

I did factor. I just didn't put it up because I was trying to understand you. And also, you were throwing out numbers without reason. & I'm glad you're not so bitter to take my words sensibly:) Thank you.