MA129 Lab Report 5 - Limits; Derivative; Derivative Rules
Fall 2021
You must be present in the Zoom lab session while completing this assignment.
1. [7 marks ] Consider the graph of f (x) given below.
(a)If possible, approximate each of the following limits.
Clearly indicate any limits approaching ±∞.
i) lim+ f (x) =
ii) lim− f (x) =
iii) lim f (x) =
iv) f (6) =
v) lim+ f (x) =
vi) lim− f (x) =
vii)lim f (x) =
viii)f (2) =
ix) lim + f (x) =
x) lim − f (x) =
xi) lim f (x) =
xii)f (−4) =
x→6
x→6
x→2
x→2
x→−4
x→−4
(b)
x→6
x→2
x→−4
i. Use your results from part (a) to discuss the continuity of f (x) at x = 2.
ii. Use your results from part (a) to discuss the continuity of f (x) at x = −4.
2. [5 marks] Find the derivative of the following functions using the limit definition. [No points will be
awarded for simply using derivative rules.]
√
(a) f (x) = −4x2 + 3
(b) f (x) = 2x + 5
3. [4 marks] Differentiate the following functions and simply the answers fully:
3
√
5x5 − 3
3
5
(b) f (x) =
(a) f (x) = (4x − 1) x + 1
−3x3 + 1
4. [6 marks] An analyst has found that a company’s costs and revenues in dollars for one product are given by
C(x) = 2x
and
R(x) = 6x −
x2
,
1000
respectively, where x is the number of items produced.
(a) Find the marginal cost function.
(b) Find the marginal revenue function.
(c) Using the fact that profit is the difference between revenue and costs, find the marginal profit function.
(d) What value of x makes marginal profit equal 0?
(e) Find the profit when the marginal profit is zero.
5. [5 marks ] Suppose you are invested in Amazon stock. Let Q(t) represent the quantity of stock you hold
at time t (measured in years). Let P (t) be the market price of Amazon stock at time t. Let V (t) be the
total value of your Amazon stock holdings at time t. Then V (t) = Q(t)P (t)
Suppose V (1) = $6300, V 0 (1) = $400, Q(1) = 7 and Q0 (1) = 0. Determine P (1) and P 0 (1) and interpret
both results.
Grade:
27