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November 01, 2024, 10:29:59 am

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Einstein_Reborn_97

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Re: Mathematics Question Thread
« Reply #4425 on: March 12, 2020, 01:20:53 pm »
0
Usually in exams, graphs are used more for detecting if points of intersection exist, and for visualising inequalities. While textbooks do this as an exercise, its important to note it down as a decent technique to see if you algebraic answer is 'in the ballpark' of where the answer should be - does it make sense? Solving for the exact values of x would of course be done algebraically :) No exam will ever ask you to take values from your graph, though some more informal exams might ask you to take approximations.

Hope this helps :)

Ok, thanks!
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Re: Mathematics Question Thread
« Reply #4426 on: April 06, 2020, 11:18:30 pm »
0
Hi Everyone! :)

I have two questions regarding the topic of the differentiation of exponential functions that I am confused with and was wondering is someone could please help me.

I have attached the questions

Thanks in advance! ;D
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fun_jirachi

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Re: Mathematics Question Thread
« Reply #4427 on: April 06, 2020, 11:31:47 pm »
+4
Hey there!

Recall that for some exponential denoted \(e^{f(x)}\), its derivative is going to be \(f'(x) \times e^{f(x)}\). Around this, we can apply things like the product rule and quotient rule - which you'll have to use for the first question!

For the second question, recall that you can find the equation of a line using \(y-y_1 = m(x-x_1)\). For some curve \(f(x)\), its tangent and normal at some point \((\alpha, f(\alpha))\) are \(y-f(\alpha) = f'(\alpha) \times (x-\alpha)\) and \(y-f(\alpha) = -\frac{1}{f'(\alpha)} \times (x-\alpha)\). Consider why this is the case, and apply it to the question!

Hope this helps :)
« Last Edit: April 09, 2020, 12:04:39 pm by fun_jirachi »
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Re: Mathematics Question Thread
« Reply #4428 on: April 09, 2020, 08:39:02 am »
0
Hey there!

Recall that for some exponential denoted \(e^{f(x)}\), its derivative is going to be \(f(x) \times e^{f'(x)}\). Around this, we can apply things like the product rule and quotient rule - which you'll have to use for the first question!

For the second question, recall that you can find the equation of a line using \(y-y_1 = m(x-x_1)\). For some curve \(f(x)\), its tangent and normal at some point \((\alpha, f(\alpha))\) are \(y-f(\alpha) = f'(\alpha) \times (x-\alpha)\) and \(y-f(\alpha) = -\frac{1}{f'(\alpha)} \times (x-\alpha)\). Consider why this is the case, and apply it to the question!

Hope this helps :)

Thankyou for the help fun_jirachi! ;D
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jasminerulez9

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Re: Mathematics Question Thread
« Reply #4429 on: April 09, 2020, 11:16:08 am »
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Could someone PLEASE help me with this question, i have been working on it for a while but just do not understand how to solve it. Help will be really appreciated, as maths is my weakest subject :(

A company finds that the function f(x)= x^3 - 96x^2 +2880x provides a good approximation for their profit f(x) in dollars, where x is the advertising expenditure in thousands of dollars

(i) What expenditure of advertising would produce the maximum profit                                                       (4 marks)

(ii) What is the maximum profit?                                                                                                                      (1 mark)

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Re: Mathematics Question Thread
« Reply #4430 on: April 09, 2020, 12:06:35 pm »
+4
Could someone PLEASE help me with this question, i have been working on it for a while but just do not understand how to solve it. Help will be really appreciated, as maths is my weakest subject :(

A company finds that the function f(x)= x^3 - 96x^2 +2880x provides a good approximation for their profit f(x) in dollars, where x is the advertising expenditure in thousands of dollars

(i) What expenditure of advertising would produce the maximum profit                                                       (4 marks)

(ii) What is the maximum profit?                                                                                                                      (1 mark)

So basically this question is asking us to find the maximum turning point on the function f(x)= x^3 - 96x^2 +2880x. So we would do this how we would normally. Differentiate the curve, find the stationary points and test to see whether they are a maximum or minimum, the maximum turning point here is the answer to (i). Part (ii) then asks us to sub in the stationary point/answer we got in part (i) into f(x) and this will give us the answer for maximum profit.

Follow these steps and it should produce the right answer! Hope this helps :)
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jasminerulez9

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Re: Mathematics Question Thread
« Reply #4431 on: April 09, 2020, 12:57:32 pm »
0
Thank you!!

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Re: Mathematics Question Thread
« Reply #4432 on: April 09, 2020, 01:38:28 pm »
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I need help with this question, I'm finding it difficult to solve. Help will be appreciated

During a survey the area of an irregular headland was to be found. The surveyor used a base line divided into ten equal sub-intervals, each of width 5m. Measurements f(x) were taken at each interval across the base line and tabulated below:

x metres  0      5      10      15      20      25      30      35      40      45
f(x)           0   10.2   13.2   16.3   13.8   16.0    17.0   18.6   10.8      0

Use the trapezoidal rule to find the approximate area of the headland    ( 5 marks)

THANK YOU!!

fun_jirachi

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Re: Mathematics Question Thread
« Reply #4433 on: April 09, 2020, 03:47:25 pm »
+3
Hey there!

With multiple applications of the trapezoidal rule on some function \(f(x)\) over some domain \(x \in [a, b]\) partitioned by x-values \(x_1, x_2 ... x_n\) such that \(a < x_1 < x_2 < ... < x_n < b\), we have that the area obtained by using the trapezoidal rule is \(\frac{h}{2}\left(f(a) + f(b) + 2(f(x_1) + f(x_2)+...+ f(x_n))\right)\), assuming, of course, that each of the partitions are equally spaced, represented by the value h \((\text{ie. }x_1 - a = x_2 - x_1 = x_3 - x_2 = ... = b - x_n = h\)) . Essentially what this means is that we have the space between each given value of x, divided by 2, multiplied by the total of the sum of the first and last terms of the function output and the sum of twice all the middle terms.

On this application, we have that \(h = 5\), and that \(f(a) = 0, f(b) = 0\). Thus, we have that the approx. area is roughly \(\frac{5}{2} (0+ 0 +2(10.2 +13.2 + 16.3 + 13.8+16.0+17.0+18.6+10.8 )) \\ = 5(115.9) = 579.5 \text{ square metres}\). Try this with any other trapezoidal rule questions you might have!

Hope this helps :)
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jasminerulez9

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Re: Mathematics Question Thread
« Reply #4434 on: April 09, 2020, 08:02:43 pm »
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Thank you soo much

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Re: Mathematics Question Thread
« Reply #4435 on: April 14, 2020, 07:12:43 pm »
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Hey Everyone,

I am confused with 2 questions and how to get to the answer, from the topic of differentiating logs.

1. The first attached image; The answer says that it is -1/x, but where did the 3 go?
2. I am generally confused with this question

Thanks in advance,
Coolmate 8)
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Re: Mathematics Question Thread
« Reply #4436 on: April 14, 2020, 08:25:21 pm »
+3
Hey Everyone,

I am confused with 2 questions and how to get to the answer, from the topic of differentiating logs.

1. The first attached image; The answer says that it is -1/x, but where did the 3 go?
2. I am generally confused with this question

Thanks in advance,
Coolmate 8)
\begin{align*}
\frac{d}{dx} (1- \ln 3x) &= -\frac{3}{3x}\\
&= -\frac1x.
\end{align*}
Simply put, the 3's cancel each other out.

For the second one, you should apply the logarithm law \( \log \frac{A}{B} = \log A - \log B\) first, to make the differentiation procedure easier.

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Re: Mathematics Question Thread
« Reply #4437 on: April 14, 2020, 10:06:10 pm »
0
\begin{align*}
\frac{d}{dx} (1- \ln 3x) &= -\frac{3}{3x}\\
&= -\frac1x.
\end{align*}
Simply put, the 3's cancel each other out.

For the second one, you should apply the logarithm law \( \log \frac{A}{B} = \log A - \log B\) first, to make the differentiation procedure easier.

Thankyou so much Rui! This makes heaps more sense now! ;D ;D
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Re: Mathematics Question Thread
« Reply #4438 on: April 23, 2020, 04:09:39 pm »
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Hey,
Could I get a hand on this question please....

Find the side a in triangleABC, where <C=60, b=24cm and the area is 30cm^2

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Re: Mathematics Question Thread
« Reply #4439 on: April 23, 2020, 10:56:56 pm »
+1
Hey there!

If we construct a perpendicular through the side AC (ie. b = 24cm) and vertex B, we have that the perpendicular is 2.5cm (0.5(24 x 2.5) = 30cm^2) - from the area of a triangle! Now, we have that 2.5/(sin 60) = BC (side a!), which is going to be \(\frac{5\sqrt{3}}{3}\) cm.

Hope this helps :)
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