Question

A Dolphin jumps and takes a path given by the equation h(t) = `1/2 (-7t^2 + 3t + 2)`, (t ≥ 0), where h(t) is the height of the Dolphin at any point in time.

1. Is the function differentiable for t ≥ 0? Justify.
2. Find the instantaneous rate of change of height at t = `1/14`.
3. h(t) is increasing in `(-∞, 3/14)`. Is this true or false? Justify.
4. Find the time at which the Dolphin will attain the maximum height. Also find the maximum height.

Answer 1

a. The given function is h(t) = `1/2 (-7t^2 + 3t + 2)`

This is a polynomial function.

Since all polynomial functions are continuous and differentiable everywhere on R, therefore, for its restricted domain t ≥ 0, it is certainly differentiable.

Yes, h(t) is differentiable for all t ≥ 0.

b. Given, t = `1/14`

h(t) = `1/2 (-7t^2 + 3t + 2)`

Differentiating w.r.t ‘t’ we get,

`(dh)/(dt) = 1/2 (-14t + 3)`

`(dh)/(dt) = -7t + 3/2`

`[(dh)/(dt)]_(t = 1/14) = -7(1/14) + 3/2`

= `-1/2 + 3/2`

= 1 unit.

c. `(dh)/(dt) = -7t + 3/2`

`(dh)/(dt) > 0`

`-7t + 3/2 > 0`

`-7t > -3/2`

`t < 3/14`

Since the domain is t ≥ 0, the interval of increase is `(0, 3/14)`.

The statement h(t) is increasing in `(-∞, 3/14)` is false, because the function is defined only for t ≥ 0.

False; h(t) is increasing only on `(0, 3/14)`.

d. Setting `(dh)/(dt)` = 0, for stationary point, we get

`-7t + 3/2 = 0`

`7t = 3/2`

`t = 3/14`

Now, `(d^2h)/(dt^2) = d/dt (-7t + 3/2)`

= −7 < 0

Since the second derivative is negative.

h(t) has a local maximum at t = `3/14`

Put the t value in the given function:

`h(3/14) = 1/2 (-7(3/14)^2 + 3(3/14) + 2)`

`h(3/14) = 1/2 (-7(9/196) + 3 (3/14) + 2)`

`h(3/14) = 1/2 (-63/196) + 9/14 + 2`

`h(3/14) = 1/2 (-9/28 + 9/14 + 2)`

`h(3/14) = 1/2 (-9/28 + 18/28 + 56/28)`

`h(3/14) = 1/2 (-9 + 18 + 56)/28`

`h(3/14) = 1/2 xx 65/28`

`h(3/14) = 65/56`

`h(3/14)` = 1.161