Calculus

Solution 1:

a)

b)

The graph break when , but the reason for the discontinuity is different , here is defined but does not exists

So is discontinuous at

And

c)

 

Solution 2:

a)

b)

c)

d)

e)

Solution 3:

a)

 

 

b)

Compute the slope of f’(4)= at x=4; m=-1/6

Find the line with slope m=-1/16 and passing through (4, ½); y=-1/16x+3/4

Solution 4:

a)

b)

c)

d)

Solution 5:

a)

We want to find f(2). We use marginal analysis to get an approximation for this value. In particular, f(2)=f(2)+f’(2)*(0)=10+7*0=10 minutes. This estimate is obtained by considering the tangent line to f at P=2. This estimate uses the T-value at P=2 on the tangent line as an approximation for f(2). Since the tangent line lies below the graph at P=2, this gives an underestimate.

b)

As the population increases, the time it takes to get to work increases at the rate of 7 minutes per 100,000 people increase in the population.

c)

We want to find f(2.5). We use marginal analysis to get an approximation for this value. In particular, f(2)=f(2)+f’(2)*(0.5)=10+7*0.5=13.5 minutes. This estimate is obtained by considering the tangent line to f at P=2.5. This estimate uses the T-value at P=2.5 on the tangent line as an approximation for f(2.5). Since the tangent line lies below the graph at P=2, this gives an underestimate.

d)

We want to find dT/dt at that point in time when P = 2 (i.e. now). We know from part b) that dT/dP = f’(2)=7 and we are told that dP/dt=0.2. Using the chain rule we get

dT/dP=(dT/dp)(dP/dt)=7*0.2=1.4 minutes/year.