HSC maths help thread (2U, MX1, MX2) (1 Viewer)

[eyeseeyou]

Given that we already have a prelim help maths thread, I decided to create this thread for you to post your HSC maths questions (for 2U, MX1 and MX2). You can use this for help or test others

I'll start: The graphs shown are of y=tanx and y=secx respectively

http://www.biology.arizona.edu/biomath/tutorials/trigonometric/graphics/trig_secant.gif
http://www.biology.arizona.edu/biomath/tutorials/trigonometric/graphics/trig_tan.gif

a) prove that secꝊ-tanꝊ=1/(secꝊ+tanꝊ)
b) Explain why 0<secꝊ-tanꝊ≤1 for 0≤Ꝋ<π/2
c) Solve the equation secꝊ-tanꝊ=1/2 for 0≤Ꝋ<π/2

(Ꝋ=0.644 radians)

(BTW this is a challenge question for the very top maths kids)

 

[eyeseeyou]

Another question:

1.a. explain why the x-coordinate of any points of the intersection of y=f(x) and y=f^-1(x) satisfies e^x - x - 4=0
b. Show the equation of e^x - x - 4=0 has the root between x=0 and x=2 and use the method of 'having by intervals' to find this root correct to the nearest number

2. a. Consider the function f(x)=e^x/(1+e^x). Find f'(x) and deduce that f(x) is increasing for all x
b. State the range of f(x)
c. Find the inverse function f^-1(x)
d. Draw y=f(x) and y=f^-1(x) on the same diagram

i am struggling to understand what these 2 Q's mean

Thanks

 

[InteGrand]

Another question:

1.a. explain why the x-coordinate of any points of the intersection of y=f(x) and y=f^-1(x) satisfies e^x - x - 4=0
b. Show the equation of e^x - x - 4=0 has the root between x=0 and x=2 and use the method of 'having by intervals' to find this root correct to the nearest number

2. a. Consider the function f(x)=e^x/(1+e^x). Find f'(x) and deduce that f(x) is increasing for all x
b. State the range of f(x)
c. Find the inverse function f^-1(x)
d. Draw y=f(x) and y=f^-1(x) on the same diagram

i am struggling to understand what these 2 Q's mean

Thanks

Which parts don't you understand? Like for question 2), a) is standard (differentiate f and show the derivative is always positive), b)'s meaning is clear, and so are the meanings of c) and d).

 

[InteGrand]

The meanings of Q.1's parts are clear too (and you haven't told us what f(x) is for that one).

 

[eyeseeyou]

For 1 the orginal question was this:

a. Shetch f(x)=e^x-4, showing clearly the coordinates of any point of intersections with the axes and the equations of any asymptotes
b. On the same diagram sketch the graph of the inverse function f^-1(x) showing clearly any important features

then there's c and d which I posted up above

 

[InteGrand]

For 1 the orginal question was this:

a. Shetch f(x)=e^x-4, showing clearly the coordinates of any point of intersections with the axes and the equations of any asymptotes
b. On the same diagram sketch the graph of the inverse function f^-1(x) showing clearly any important features

then there's c and d which I posted up above

You do understand what the Q's mean though, right?

 

[eyeseeyou]

Another question:

1.a. explain why the x-coordinate of any points of the intersection of y=f(x) and y=f^-1(x) satisfies e^x - x - 4=0
b. Show the equation of e^x - x - 4=0 has the root between x=0 and x=2 and use the method of 'having by intervals' to find this root correct to the nearest number

2. a. Consider the function f(x)=e^x/(1+e^x). Find f'(x) and deduce that f(x) is increasing for all x
b. State the range of f(x)
c. Find the inverse function f^-1(x)
d. Draw y=f(x) and y=f^-1(x) on the same diagram

i am struggling to understand what these 2 Q's mean

Thanks

This you solve simultaneously (bolded)

This I believe is Integration (underlined)

Please correct me if I am wrong

 

[InteGrand]

This you solve simultaneously (bolded)

This I believe is Integration (underlined)

Please correct me if I am wrong

 

[eyeseeyou]

So you just sub in the values? (sob the x value so then you get a y value)

Thanks

 

[InteGrand]

So you just sub in the values? (sob the x value so then you get a y value)

Thanks

Yes, and show and state that they're of opposite sign.

 

[eyeseeyou]

Yes, and show and state that they're of opposite sign.

Thanks

 

[eyeseeyou]

Another question:

1.a. explain why the x-coordinate of any points of the intersection of y=f(x) and y=f^-1(x) satisfies e^x - x - 4=0
b. Show the equation of e^x - x - 4=0 has the root between x=0 and x=2 and use the method of 'having by intervals' to find this root correct to the nearest number

2. a. Consider the function f(x)=e^x/(1+e^x). Find f'(x) and deduce that f(x) is increasing for all x
b. State the range of f(x)
c. Find the inverse function f^-1(x)
d. Draw y=f(x) and y=f^-1(x) on the same diagram

i am struggling to understand what these 2 Q's mean

Thanks

For this example, I am confused because it's an exponential over and exponential

 

[InteGrand]

For this example, I am confused because it's an exponential over and exponential

Well differentiating that is still just using the quotient rule. Have you tried using the quotient rule?

 

[eyeseeyou]

Well differentiating that is still just using the quotient rule. Have you tried using the quotient rule?

Yeah I know you have to do e^x/(1+e^x)>0

I haven't tried it (and in this case I don't really know how to use the quotient rule for exponentials)

Also what is this rule called:

y=3(4x+1)^2
dy/dx=24(4x+1)

 

[InteGrand]

Also what is this rule called:

y=3(4x+1)^2
dy/dx=24(4x+1)

That is the chain rule.

 

[InteGrand]

Yeah I know you have to do e^x/(1+e^x)>0

Which part are you referring to for this? The range? That's not sufficient to get the range.

 

[eyeseeyou]

It said deduce f(x) is increasing for all x so doesn't that mean e^x/(1+e^x)>0 (must be the case as it is increasing)

 

[InteGrand]

Then just simplify that.

 

[eyeseeyou]

Then just simplify that.

Thank you

 

[InteGrand]

It said deduce f(x) is increasing for all x so doesn't that mean e^x/(1+e^x)>0 (must be the case as it is increasing)

No, that's irrelevant. E.g. The function g(x) = -e^-x is increasing, yet always negative.