SmarterEd

Aussie Maths & Science Teachers: Save your time with SmarterEd

Calculus, MET2 2021 VCAA 19 MC

Which one of the following functions is differentiable for all real values of `x`?

  1. `f(x) = {{:(qquadx),(–x):} {:(qquadx<0),(qquadx≥0):}`
  2.  `f(x) = {{:(qquadx),(–x):} {:(qquadx<0),(qquadx>0):}`
  3.  `f(x) = {{:(8x + 4),((2x + 1)^2):} {:(qquadx<0),(qquadx≥0):}`
  4.  `f(x) = {{:(2x + 1),((2x + 1)^2):} {:(qquadx<0),(qquadx≥0):}`
  5.  `f(x) = {{:(4x + 1),((2x + 1)^2):} {:(qquadx<0),(qquadx≥0):}`
Show Answers Only

`E`

Show Worked Solution

`text{Differentiable → function must be continuous and smooth}`

♦♦ Mean mark 35%.

`text{Consider}\ E:`
 

`f(x) = {{:(4x + 1),((2x + 1)^2):} {:(qquadx<0),(qquadx≥0):}  \ => \ f′(x) = {{:(4),(4(2x + 1)):} {:(qquadx<0),(qquadx≥0):}`
 

`text{As} \ x to 0^- \ , \ f(x) to \ 1 \ , \ f′(x) = 4`

`text{As} \ x to 0^+ \ , \ f(x) to \ 1 \ , \ lim f′(x) = 4`

`:. \ text{Function is continuous and smooth}`

`=> E`