Question

Let f and g be the functions defined by f(t) = 2t2 and g(t) = t3 + 4t.

1) Determine f'(t) and g′(t).
2) Let p(t) = 2t2 (t3 + 4t) and observe that p(t) = f(t) ⋅ g(t). Re-write the formula for p by distributing the 2t2 term. Then, compute p′(t) using the sum and constant multiple rules.
3) p′(t) = f′(t) ⋅ g′(t).
A. True
B. False
4) Let q(t) = t3 + 4t2/2t2 and observe that q(t) = g(t)/f(t). Rewrite the formula for q by dividing each term in the numerator by the denominator and simplify to write q as a sum of constant multiples of powers of t. Then, compute q′(t) using the sum and constant multiple rules.
5) q′(t) = g′(t)/f'(t).
A. True
B. False

Answer 1

1)
`f'(t)=4t,\ g'(t)=3t^2+4`

2)
`p(t) =2t^5+8t^3`

`p'(t)=10t^4+24t^2`

3) False

4)
`q(t) =(1)/(2)t+2t^(-1)`

`q'(t)=(1)/(2)-(2)/(t^2)`

5) False

Explanation:

Given that:

`f(t) = 2t^2` and
`g(t) = t^3 + 4t`

Formula:

`1. (d)/(dx)x^n=nx^(n-1)`

`2. (d)/(dx)C.f(x)=C.f'(x)\ \{\text{C is a constant}\}`

1) Using above formula:

`f'(t)=2* 2 t^(2-1)=4t`

`g'(t)=3t^(3-1)+4* 1 t^(1-1)=3t^2+4`

2)
`p(t) =2t^2(t^3+4t)`

Rewriting the formula by distributing the
`2t^2` term:

`p(t) =2t^2.t^3+2t^2.4t=2t^5+8t^3`

`p'(t) = 10t^4+24t^2`

3) By using answers of part (1):

`f'(t).g'(t)=12t^3+16t`

`p'(t) = 10t^4+24t^2`

Therefore it is False that
`p'(t) = f'(t).g'(t)`

4)
`q(t)=(t^3+4t)/(2t^2)`

Writing by distributing:

`q(t)=(t^3)/(2t^2)+(4t)/(2t^2)\\\Rightarrow q(t) =(t)/(2)+(2)/(t)\\\Rightarrow q(t) =(1)/(2)t+2t^(-1)`

Using the formula:

`q'(t)=(1)/(2)t^(1-1)+2(-1)/(t^2)\\\Rightarrow q'(t)=(1)/(2)-(2)/(t^2)`

(5)By using answers in part (1):

`(g'(t))/(f'(t))=(3t^2+4)/(4t)=(3)/(4)t+\frac{1}t`

`q'(t)=(1)/(2)-(2)/(t^2)`

Therefore, it is False that:

`q'(t)=(g'(t))/(f'(t))`