Math123

MATH123 (2012) D1 Assignment 2 Solutions. Calculus 1. Find the derivatives of the following functions using the various rules for computing derivatives: (a) f (x) = 2x2 + 1 √ (b) g(x) = x − x (c) r(t) = (t2 + t)(5 − t) √ (d) q(s) = (s4 + 2s3 − 1)6

√

(e) w(z) =

z+1 z−1

Solution

(a) f (x) = 4x (b) f (x) = x −

1 √ 2 x

(c) We could use the product rule here, but it is perhaps easier not to. First r(t) = 5t2 − t3 + 5t − t2 So r (t) = 10t − 3t2 + 5 − 2t. (d) Using the chain rule q (s) = 6(s4 + (e) By the quotient rule w (z) =

(z−1) √ 2 z+1

√

√ 2s3 − 1)5 (4s3 + 3 2s2 )

−

√

z+1

(z − 1)2 −z − 3 = √ 2 z + 1(z − 1)2

2. Consider the function y = f (x) where f (x) = 2x4 − x3 − 7x2 − x (a) Find the values of f (−2), f (−1), f (0), f (1) and f (2) (b) Compute f (x), and evaluate f (−2), f (−1), f (0), f (1) and f (2). (c) Draw a graph of y = f (x) which is consistent with this information.

Solution

(a) f (−2) = 14, f (−1) = −3, f (0) = 0, f (1) = −7, f (2) = −6. (b) f (x) = 8x3 − 3x2 − 14x − 1, so f (−2) = 49, f (−1) = 2, f (0) = −1, f (1) = −10, f (2) = 23.

(c)

3. Lamborghini recently introduced its new Aventador sports car, and its manufacturing team has put together a schedule for production. This schedule involves starting by producing a small number of cars R(t) per month t while manufacturing and quality control issues are worked out, and then increasing production rapidly over a period of 4 years. The companies financial division has also predicted the profitability P (r), which gives the rate of profit generation, in dollars per month as a function of r, the monthly production of Aventadors. The plan is to begin with production of 4 cars per month for the first 4 months, rising to a total of 20 cars per month, with a dramatic scaling up of production roughly between month 10 and month 30. Production will be fixed at 20 cars per month in months 44 onwards. This can be seen in the graph below:

The financial projections...