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Calculus, SPEC1 2020 VCAA 7

Consider the function defined by

`f(x) = {({:mx + n,:}, x < 1), ({: frac{4}{(1 + x^2)},:}, x >= 1):}`

where `m` and `n` are real numbers.

  1. Given that  `f(x)`  and  `f^{′}(x)`  are continuous over  `R`, show that  `m = -2`  and  `n = 4`.  (2 marks)

  2. Find the area enclosed by the graph of the function, the x-axis and the lines  `x = 0`  and  `x = sqrt 3`.  (3 marks)

Show Answers Only

  1. `text(Proof)\ text{(See Worked Solutions)}`
  2. `3 + pi/3\ text(u²)`

Show Worked Solution

a.  `f(x)\ \ text(continuous when:)`

`m(1) + n` `= 4/(1 + 12)`
`m + n` `= 2`

 
`f^{′} (x) = {({:m,:}, x < 1), ({:(-8x)/(1 + x^2),:}, x >= 1):}`
 

`f(x)\ \ text(continuous when:)`

`m = (-8(1))/(1+1^2)^2 = -2`

`text(When)\ \ m =-2,\ n = 4`

 

b.   `f(x) = {({:4-2x,:}, x < 1), ({:4/(1 + x^2),:}, x >= 1):}`

`f(x) > 0\ \ text(for)\ \ x ∈ [0, sqrt 3]`

`A` `= int_1^sqrt 3 4/(1 + x^2)\ dx + int_0^1 4-2x\ dx`
  `= [4 tan^(-1)(x)]_1^sqrt 3 + [4x-x^2]_0^1`
  `= (4 pi)/3-(4 pi)/4 + 4-1`
  `= 3 + pi/3\ text(u²)`

Calculus, SPEC2 2011 VCAA 14 MC

If  `f″(x) = 2e^xsin(x)`,  `f′(0) = 0`  and  `f(0) = 0`, the  `f(x)`  equals

A.   `−e^x(cos(x) + sin(x))`

B.   `−e^x(cos(x) - sin(x)) + 1`

C.   `−e^xcos(x)`

D.   `x - e^xcos(x) + 1`

E.   `x - e^xcos(x)`

Show Answers Only

`D`

Show Worked Solution

`text(Consider:)\ \ f(0)=0,`

`=>\ text(Eliminate A, C and E.)`

`text(By CAS, B and D satisfy)\ \ f′(0) = 0.`

`text(By CAS, only D satisfies)\ \ f″(x) = 2e^x sin(x)`

`=> D`