Question

I need help in Calculus Differentiation. Attachment inside

Answer 1

specifically 9 and 10

Answer 2

lets start from number 1, k>?

Answer 3

or you got the simple derivative questions?

Answer 4

i understand 1-3 and 5 and 6

Answer 5

so lets go from #4

Answer 6

alright thanks

Answer 7

f'(a) is your slope, of a tangent line to the (explicit) function, at x=a

Answer 8

apply the power rule to each term to find the derivative of the function.

Answer 9

what do you get for the f'(x) ?

Answer 10

4x^2-12

Answer 11

4x^2? are you sure?

Answer 12

4x - 12

Answer 13

yes, f'(x)=4x-12

Answer 14

now, your line is supposed to be tangent at a point (5,9).
So find f'(5) for me (plug in 5 instead of x, into the derivative)

Answer 15

(f'(5) will be your slope)

Answer 16

8

Answer 17

yes, f'(5)=8

Answer 18

So your slope is 8. your point is (5,9)

use: \(\large\color{slate}{ y-y_1=m(x-x_1) }\) to find your line

Answer 19

y-9=8(x-5)

Answer 20

yes.

Answer 21

If you are required to simplify to y-intercept form, then do so... if not, then whatever you want...

as you wish and according to the teachers requirements.

Answer 22

We can move on to number 7, okay?

Answer 23

thanks

Answer 24

okay

Answer 25

oh, wait you said you don't get #2?

Answer 26

oh wait i have 7

Answer 27

i do get 2

Answer 28

okay, lets do number 2.

will start from b. \(\large\color{slate}{ y=x^{~-1/3} +5x^{2/3} }\)

Answer 29

apply the power rule to each term, to find the derivative.

Answer 30

no wait i said i do understand 2

Answer 31

oh, so you don't need help on a or b, in number 2?

Answer 32

no sorry for the confusion

Answer 33

sure, so you need 7 or 8?

Answer 34

8, correct?

Answer 35

8

Answer 36

Okay, \(\large\color{slate}{ y=x^3-10x^2+12x+23 }\) over the interval \(\large\color{slate}{ [-1,3] }\).

Answer 37

Is that a correct interpretation, or is it an open interval (which would be more "between" x=-1 and x=3) ?

Answer 38

im not sure

Answer 39

okay, lets find the derivative first.

Answer 40

\(\large\color{slate}{ y=x^3-10x^2+12x+23 }\) (apply power rule to each term)

differentiate for me please.

Answer 41

3x^2-20x+12

Answer 42

yes, that is right:) \(\large\color{slate}{f'(x)=3x^2-20x+12 }\)

Answer 43

set \(\large\color{slate}{f'(x)=0 }\) to solve for the critical numbers

Answer 44

I mean like this, \(\large\color{slate}{0=3x^2-20x+12 }\)

Answer 45

one second

Answer 46

(yes, it is factorable)

Answer 47

sure, take your time

Answer 48

x=2/3 ans 6

Answer 49

*and

Answer 50

yes, 2/3 and 6

Answer 51

So, your function is: \(\large\color{slate}{f(x)=x^3-10x^2+12x+23 }\)

please, find:
~ \(\large\color{slate}{f(\frac{2}{3}) }\)

~ \(\large\color{slate}{f(6) }\)

Answer 52

so 26.85 and -49

Answer 53

I have checked, I think it doesn't matter whether the interval [-1,3] is open or not in this case.

since you need the greatest output, and that is not when x=-1, or x=3

Answer 54

I don't think that it is correct.

Answer 55

\(\large\color{slate}{f(2/3) }\)
http://www.wolframalpha.com/input/?i=%282%2F3%29%5E3-10%282%2F3%29%5E2%2B%282%2F3%29%2B23

\(\large\color{slate}{f(6) }\)
http://www.wolframalpha.com/input/?i=%286%29%5E3-10%286%29%5E2%2B%286%29%2B23

Answer 56

i think your missing the plus 12

Answer 57

oh, true that

Answer 58

\(\large\color{slate}{f(2/3) }\)
http://www.wolframalpha.com/input/?i=%282%2F3%29%5E3-10%282%2F3%29%5E2%2B12%282%2F3%29%2B23
\(\large\color{slate}{26.85 }\)

\(\large\color{slate}{f(6) }\)
http://www.wolframalpha.com/input/?i=%282%2F3%29%5E3-10%282%2F3%29%5E2%2B12%282%2F3%29%2B23
\(\large\color{slate}{-49 }\)

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

\(\large\color{slate}{f(-1) }\)
http://www.wolframalpha.com/input/?i=%282%2F3%29%5E3-10%282%2F3%29%5E2%2B12%282%2F3%29%2B23
\(\large\color{slate}{-1 }\)

\(\large\color{slate}{f(3) }\)
http://www.wolframalpha.com/input/?i=%282%2F3%29%5E3-10%282%2F3%29%5E2%2B12%282%2F3%29%2B23
\(\large\color{slate}{-4 }\)

Answer 59

yes, but as again, whether, the interval of "from -1 to 3" is open or closed, that doesn't matter

Answer 60

did you also want me to do f(-1) and f(3) ?

Answer 61

I just checked, if `"between x=-1 and x=3"` would make a difference if it is an open interval or closed.
Saying if x=-1,3 are or aren't included

Answer 62

So your answer for absolute maximum would be:
\(\large\color{slate}{(\frac{2}{3},26.85) }\)

because with x=2/3, you get the biggest output

Answer 63

I mean the point, not the absolute maximum.

Answer 64

the point where the absolute maximum on a function occurs. (rephrasing)

Answer 65

so the coordinates are (2/3,26.85) and (6,-49)

Answer 66

yes, but you need the point where the maximum is.
And f(6) is a minimum

Answer 67

So, you would just have \(\large\color{slate}{(\color{red}{\frac{2}{3}},26.85) }\)

Answer 68

(as your answer)

Answer 69

Need more clarification with number 8, or we can move on to 9?

Answer 70

we can move on

Answer 71

My connection is bad, apologize.

Answer 72

\(\large\color{slate}{y= 4x^3-24x^2+45x-23 }\) differentiate this, for me please

Answer 73

12x-48+45

Answer 74

yes, that is correct.

Answer 75

\(\large\color{slate}{f'(x)= 12x^2-48x+45 }\). Now, you need to find
the slope of the tangent.

your tangent, is tangent at (1,2)

so find \(\large\color{slate}{f'(1) }\) for me real quick....

Answer 76

(plug in 1, instead of x, into the derivative)

Answer 77

to the original right

Answer 78

no, into the \(\large\color{slate}{f'(x) }\)

Answer 79

9

Answer 80

yes

Answer 81

so your slope is 9.

Answer 82

now, please find the entire line, knowing that the point is \(\large\color{slate}{(1,2) }\), and the slope is \(\large\color{slate}{9 }\).

Answer 83

y=9x-7

Answer 84

yes

Answer 85

that is correct.

Answer 86

@SolomonZelman are you sure about the points? when the slope is 0 at the x intercept (2/3, 0) no and the value of y = -26

Answer 87

I am not sure what you are trying to tell me

Answer 88

I am asking if you are sure about
"so the coordinates are (2/3,26.85) and (6,-49)"

Answer 89

I never said that.
I said that the absolute maximum is 26.85 and occurs at x=2/3

Answer 90

(6,-49) is the absolute minimum.

Answer 91

but the absolute minimum, as I have previously said, is not what we needed. the answer for number 8 was just (2/3, 26.85) >> the absolute maximum point

Answer 92

-6 is not the minimum, axcuse me mam

Answer 93

it is not in the interval... but that is ain't important for the problem.

Answer 94

nice that you caught me though

Answer 95

so the coordinates are (2/3,26.85) and (6,-49)

10 minutes ago

99
SolomonZelman Honorary Professor of Mathematics
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yes, but you need the point where the maximum is. And f(6) is a minimum