This discussion is now closed.

Check out other Related discussions

- Edexcel Past Papers
- Edexcel A2 Mathematics: Core Mathematics C4 6666 01 - 22 June 2018 [Exam Discussion]
- Further maths specimen papers?!
- Edexcel Maths AS+A2 - All papers in one excel sheet (including solomon papers)!
- Do they make mocks harder than thee real GCSE exams?
- Business AS
- IGCSE English language B ❤️😭 23 May 2023
- 1000+ A2-Level Biology Exam Questions
- GCSE Computer Science Study Group 2023-2024
- GCSE Geography (Edexcel B)
- Help gcse business
- Is it possible to go from a grade 4 to a 7 or higher in GCSE Maths?
- WJEC A-Level Geology 2024
- Specimen papers for edexcel english lit
- IGCSE English Language B Section C
- CB3 paper
- NOTES FOR GCSE URGENT + gimme some tips for grade 9 pls
- Important about gcse
- english language paper 1 question 5
- Large Data Set A level Maths

Gaz031

What's the question?

Erm ok the first part is:

Given that y = tan x + 2 cos x, find the exact value of dy/dx at x = pi/4

.. I differentiated it to dy/dx = sec^2x - 2 sin x

but then i get confused with substituting pi/4 in place of x cos it wants the exact value

And also the answer is 2 - square root of 2 (if it helps!)

violet

Erm ok the first part is:

Given that y = tan x + 2 cos x, find the exact value of dy/dx at x = pi/4

.. I differentiated it to dy/dx = sec^2x - 2 sin x

but then i get confused with substituting pi/4 in place of x cos it wants the exact value

And also the answer is 2 - square root of 2 (if it helps!)

Given that y = tan x + 2 cos x, find the exact value of dy/dx at x = pi/4

.. I differentiated it to dy/dx = sec^2x - 2 sin x

but then i get confused with substituting pi/4 in place of x cos it wants the exact value

And also the answer is 2 - square root of 2 (if it helps!)

So you want (sec pi/4)^2 - 2sin(pi/4) = 1/(cos pi/4)^2 - 2sin(pi/4) = 2 - 2[0.5rt2] = 2-rt2.

If ever they want an exact value and you can't immediately recall it (eg value of sin pi/4) try putting it in on your calculator and squaring the number to see if you have a surd or dividing by constants such as pi.

7 (i) Given that y = tan x + 2 cos x, find the exact value of dy/dx at x = pi/4

dy/dx = sec^2x - 2sinx

At x = pi/4,

dy/dx = 1/(cos(pi/4))^2 - 2sin(pi/4)

= 1/(1/root2)^2 - 2root2/2

= 1/(1/2) - (2root2)/2

= 2 - root2

(ii) Given that x = tan 0.5y, prove that dy/dx = 2/(1+x^2)

dx/dy = 0.5sec^2 0.5y

dy/dx = 1/(dx/dy) = 1/(0.5sec^2 0.5y) = 2/sec^2 0.5y

sec^2(A) = 1 + tan^2(A)

dy/dx = 2/(1 + tan^2 0.5y) = 2/(1 + x^2)

dy/dx = sec^2x - 2sinx

At x = pi/4,

dy/dx = 1/(cos(pi/4))^2 - 2sin(pi/4)

= 1/(1/root2)^2 - 2root2/2

= 1/(1/2) - (2root2)/2

= 2 - root2

(ii) Given that x = tan 0.5y, prove that dy/dx = 2/(1+x^2)

dx/dy = 0.5sec^2 0.5y

dy/dx = 1/(dx/dy) = 1/(0.5sec^2 0.5y) = 2/sec^2 0.5y

sec^2(A) = 1 + tan^2(A)

dy/dx = 2/(1 + tan^2 0.5y) = 2/(1 + x^2)

violet

Ok yeh i get the first bit but i dont get the root 2 (sorry to be really annoying) but i thought sin (pi/4) is 1/√2 so when you multiply it by 2 doesn't it become 2/√2 ? Or am i totally missing the point here..

Rationalise the denominator.

E.g. 1/√2 = 1/√2 x √2/√2 <--- (effectively multiplying by 1 here) = (√2)/2

Try both 1/√2 and(√2)/2 on your calculator.

so 2sin(pi/4) = 2 x (√2)/2 = √2

Gaz031

For (iii) you use the product rule to differentiate, then pull out e^-x so you have dy/dx=e^-x[acos2x+bsin2x]. You then express acos2x+bsin2x in the form Rcos(2x+a) by setting acos2x+bsin2x=Rcos(2x+a) and expanding out Rcos(2x+a) then equating coefficients of cos2x and sin2x.

Yep I've done the differentiating, I just don't get how i can express it in that form. Where does the sin 2x go?

(iii) Given that y = e^–x sin 2x, show that dy/dx can be expressed in the form Re^–x cos (2x + @). Find, to 3 significant figures, the values of R and α, where 0 < α < pi/2

y = e^-x sin2x

u = e^-x, v = sin2x

dy/dx = u*dv/dx + v*du/dx (product rule)

dy/dx = e^-x(2cos2x) + sin2x(-e^-x)

dy/dx = 2e^-x(cos2x) - e^-x(sin2x)

dy/dx = e^-x(2cos2x - sin2x)

= Re^-x cos(2x + @) [cos(A+B)]

= Re^-x(cos2xcos@ - sin2xsin@)

= Re^-x(cos2xcos@) - Re^-x(sin2xsin@)

Comparing coefficients gives us.

Rcos@ = 2 and Rsin@ = 1

(Rcos@)^2 + (Rsin@)^2 = 2^2 + 1^2 = 5 = R^2

so R = rt5 = 2.24

Rcos@ = 2

cos@ = 2/rt5

@ = cos^-1(2/rt5) = 0.464

y = e^-x sin2x

u = e^-x, v = sin2x

dy/dx = u*dv/dx + v*du/dx (product rule)

dy/dx = e^-x(2cos2x) + sin2x(-e^-x)

dy/dx = 2e^-x(cos2x) - e^-x(sin2x)

dy/dx = e^-x(2cos2x - sin2x)

= Re^-x cos(2x + @) [cos(A+B)]

= Re^-x(cos2xcos@ - sin2xsin@)

= Re^-x(cos2xcos@) - Re^-x(sin2xsin@)

Comparing coefficients gives us.

Rcos@ = 2 and Rsin@ = 1

(Rcos@)^2 + (Rsin@)^2 = 2^2 + 1^2 = 5 = R^2

so R = rt5 = 2.24

Rcos@ = 2

cos@ = 2/rt5

@ = cos^-1(2/rt5) = 0.464

y=(e^-x)sin2x

dy/dx=(e^-x)(2cos2x)+(-e^-x)sin2x

dy/dx=(e^-x)[2cos2x-sin2x]

Let Rcos(2x+a)=2cos2x-sin2x

R[cos2xcosa-sin2xsina]=2cos2x-sin2x

cos2x[Rcosa]-sin2x[Rsina]=2cos2x-sin2x.

Rcosa=2, Rsina=1.

tana=Rsina/Rcosa=1/2, hence a=arctan0.5.

R^2 = 1^2 + 2^2 = 5, hence R=rt5.

dy/dx=(rt5)(e^-x)cos(2x+arctan0.5)

Use your calculator to give R and a to the required accuracy.

dy/dx=(e^-x)(2cos2x)+(-e^-x)sin2x

dy/dx=(e^-x)[2cos2x-sin2x]

Let Rcos(2x+a)=2cos2x-sin2x

R[cos2xcosa-sin2xsina]=2cos2x-sin2x

cos2x[Rcosa]-sin2x[Rsina]=2cos2x-sin2x.

Rcosa=2, Rsina=1.

tana=Rsina/Rcosa=1/2, hence a=arctan0.5.

R^2 = 1^2 + 2^2 = 5, hence R=rt5.

dy/dx=(rt5)(e^-x)cos(2x+arctan0.5)

Use your calculator to give R and a to the required accuracy.

Gaz031

y=(e^-x)sin2x

dy/dx=(e^-x)(2cos2x)+(-e^-x)sin2x

dy/dx=(e^-x)[2cos2x-sin2x]

Let Rcos(2x+a)=2cos2x-sin2x

R[cos2xcosa-sin2xsina]=2cos2x-sin2x

cos2x[Rcosa]-sin2x[Rsina]=2cos2x-sin2x.

Rcosa=2, Rsina=1.

tana=Rsina/Rcosa=1/2, hence a=arctan0.5.

R^2 = 1^2 + 2^2 = 5, hence R=rt5.

dy/dx=(rt5)(e^-x)cos(2x+arctan0.5)

Use your calculator to give R and a to the required accuracy.

dy/dx=(e^-x)(2cos2x)+(-e^-x)sin2x

dy/dx=(e^-x)[2cos2x-sin2x]

Let Rcos(2x+a)=2cos2x-sin2x

R[cos2xcosa-sin2xsina]=2cos2x-sin2x

cos2x[Rcosa]-sin2x[Rsina]=2cos2x-sin2x.

Rcosa=2, Rsina=1.

tana=Rsina/Rcosa=1/2, hence a=arctan0.5.

R^2 = 1^2 + 2^2 = 5, hence R=rt5.

dy/dx=(rt5)(e^-x)cos(2x+arctan0.5)

Use your calculator to give R and a to the required accuracy.

nice.

I have another question... It's a little one though!

You see for question 2(c) on that same paper.. Well what do you do after you've done f(2.09455) and f(2.09465) ..(the upper and lower limits) .. Do you just say that's its correct because there is a change of sign between the 2 answers? Or do you have to do another little calculation?

You see for question 2(c) on that same paper.. Well what do you do after you've done f(2.09455) and f(2.09465) ..(the upper and lower limits) .. Do you just say that's its correct because there is a change of sign between the 2 answers? Or do you have to do another little calculation?

violet

I have another question... It's a little one though!

You see for question 2(c) on that same paper.. Well what do you do after you've done f(2.09455) and f(2.09465) ..(the upper and lower limits) .. Do you just say that's its correct because there is a change of sign between the 2 answers? Or do you have to do another little calculation?

You see for question 2(c) on that same paper.. Well what do you do after you've done f(2.09455) and f(2.09465) ..(the upper and lower limits) .. Do you just say that's its correct because there is a change of sign between the 2 answers? Or do you have to do another little calculation?

You might also have to say f is continuous in the interval so the root is in the interval which is 2.0946 (correct to 4dp)

- Edexcel Past Papers
- Edexcel A2 Mathematics: Core Mathematics C4 6666 01 - 22 June 2018 [Exam Discussion]
- Further maths specimen papers?!
- Edexcel Maths AS+A2 - All papers in one excel sheet (including solomon papers)!
- Do they make mocks harder than thee real GCSE exams?
- Business AS
- IGCSE English language B ❤️😭 23 May 2023
- 1000+ A2-Level Biology Exam Questions
- GCSE Computer Science Study Group 2023-2024
- GCSE Geography (Edexcel B)
- Help gcse business
- Is it possible to go from a grade 4 to a 7 or higher in GCSE Maths?
- WJEC A-Level Geology 2024
- Specimen papers for edexcel english lit
- IGCSE English Language B Section C
- CB3 paper
- NOTES FOR GCSE URGENT + gimme some tips for grade 9 pls
- Important about gcse
- english language paper 1 question 5
- Large Data Set A level Maths

Latest

Trending