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MATH 1410SEF & S141F(W) Algebra and Calculus

Assignment 2

Due: 5 Nov 2023 (Sun) 23:59

The full mark of this assignment is 100 marks. Please submit your answers with clear steps in a single PDF file to the OLE. You should only hand-write your answers (on papers or tablet).

Question 1 (14 marks). Evaluate  f(x) for each of the following.

Question 2 (10 marks). Use the first principles of derivatives to find the derivative of 

Question 3 (15 marks). Let k be a constant. Consider

(a) Find g(3).

(b) Find  g(x).

(c) If g is a continuous function, find the value of k.

Question 4 (21 marks). Find the derivatives of the following with respect to x.

Question 5 (10 marks). Let y = 5x 3 + ln(4x) + 9. Evaluate 

Question 6 (10 marks). Find  in terms of x and/or y, where 3x2 + 5y = xy.

Question 7 (10 marks). Use logarithmic differentiation to find the derivative of y = 2x−2x 2 .

Question 8 (10 marks) [Profit and Marginal profit]. Let P(x) be the profit from producing (and selling) x units of goods. Then, the marginal profit function is defined to be the first derivative P ′ (x) of the profit function P(x). Match each of the following questions with the proper solution.

(a) What is the profit from producing 500 units of goods?

(b) At what level of production will the marginal profit be 500 dollars?

(c) What is the marginal profit from producing 500 units of goods?

(d) For what level of production will the profit be 500 dollars?

Solution options:

(I) Compute P′ (500).

(II) Find a value of a for which P′ (a) = 500.

(III) Set P(x) = 500 and solve for x.

(IV) Compute P(500).