please show me how to do them

Question

Answer 1

(15)

The height of the cliff is s(t) = -16t² +64t +200 .

A ball is thrown upwards from the cliff .

So the initial position of the ball is s(t) = -16t² +64t +200 .

The velocity of the ball can be measured by find the first derivative of its position .

v(t) = s' (t)

v(t) = -32t + 64

To find the time when the velocity of ball is zero , make the velocity function equate to zero .

 -32t + 64 = 0

32t = 64

t = 2 .

So at the time t = 2 sec the velocity of ball is zero .

The height of the ball at t = 2 sec is 

s(2) = -16 (2²) +64(2) +200

s(2) = - 64  + 128 + 200

s(2) = 264 

So the height of the ball when velocity equal to zero is 264 feet .

Answer 2

(14)

To estimate the smallest value of derivative of f(x) at given points , we make use of maximum and minimum concept .

The function f(x) has a maximum / minimum  value when f '(x) = 0 . 

when the function f(x) increases then f'(x) will decrease and vice versa .

The graph of f(x) has given , make rough graph of f '(x) .

 

So the smallest value of f'(x) in the given points is at point D .

So option D is correct .