The consumption of tungsten (in metric tons) in a country is given approximately by

p(t)=138t2+1,071t+14,910,

where t is time in years and

t=0

corresponds to 2010.(A) Use the four-step process to find

p(t).

(B) Find the annual consumption in

2028

and the instantaneous rate of change of consumption in

2028,

and write a brief verbal interpretation of these results.

example-

The consumption of tungsten (in metric tons) in a country is given approximately by

p(t)=136t2+1,072t+14,905,

where t is time in years and

t=0

corresponds to 2010.

(A) Use the four-step process to find

p(t).

(B) Find the annual consumption in

2025

and the instantaneous rate of change of consumption in

2025,

and write a brief verbal interpretation of these results.

(A) For

y=f(x),

the derivative of f at x, denoted

f(x),

is defined by the formula below, provided the limit exists.

f(x)=limh0f(x+h)f(x)h

First find

p(t+h).

Substitute

t+h

for t in the formula for p(t).

p(t+h)

=

136(t+h)2+1,072(t+h)+14,905

Now determine

p(t+h)p(t).

p(t+h)p(t)	=	136(t+h)2+1,072(t+h)+14,905136t2+1,072t+14,905
	=	136t2+2ht+h2+1,072(t+h)+14,905136t2+1,072t+14,905
	=	136t2+272ht+136h2+1,072t+1,072h+14,905136t21,072t14,905
	=	272ht+136h2+1,072h

Next find

p(t+h)p(t)h.

p(t+h)p(t)h	=	272ht+136h2+1,072hh
	=	h(272t+136h+1,072)h	Factor h out of the numerator.
	=	272t+136h+1,072	Simplify.

Finally, determine

p(t)

by finding the limit of

p(t+h)p(t)h

as h approaches 0. Notice that since

272t+136h+1,072

is a polynomial, the limit is the same as evaluating the expression at

h=0.

p(t)	=	limh0 p(t+h)p(t)h
	=	limh0 (272t+136h+1,072)
	=	272t+1,072	Calculate the limit.

Therefore,

p(t)=272t+1,072.

(B) To find the annual consumption in

2025

and the instantaneous rate of change of consumption in

2025,

first determine the t-value corresponding to year

2025.

t=15

To find the annual consumption in

2025,

evaluate the function

p(t)=136t2+1,072t+14,905

at

t=15.

p(t)	=	136t2+1,072t+14,905
p(15)	=	136(15)2+1,072(15)+14,905	Substitute 15 for t.
	=	61,585	Simplify.

To find the instantaneous rate of change of consumption in

2025,

evaluate the function

p(t)=272t+1,072

at

t=15.

p(t)	=	272t+1,072
P(15)	=	272(15)+1,072	Substitute 15 for t.
	=	5,152	Simplify.

Therefore, the annual consumption in

2025

is

61,585

metric tons and the instantaneous rate of change of consumption in

2025

is

5,152

metric tons.

This means that in

2025,

61,585

metric tons of tungsten are consumed and this quantity is increasing at the rate of

5,152

metric tons per year.