BUSINESS CALCULUS INCOURSE TEST #1 (UNITS 1 & 2) SEMESTER 2 2022 SECTION A Instructions: Attempt ALL questions in this section. 1. Let f(x) be a function; what does it mean to say "f(x) has a limit at x = a"? A. The function is defined at x = a. B. The left-hand limit exists at x = a. C. The right-hand limit exists x = a. D. The left-hand limit and right-hand limit exist at x = a and are both equal. 2. Consider the following graph. What is the equation of the asymptote present? A. y= 3 B. x=3 C. y= = D. y = - 1_1 x 2 Questions 3-4 depends on the following graph of the arbitrary function y = f(x).2. Consider the following graph. What is the equation of the asymptote present? A. y=3 B. x=3 C. y= D. y = - 1 Questions 3-4 depends on the following graph of the arbitrary function y = fix). .2 2 3 4 5 6 3. What is the value of f(x) when x = 3? A. The graph is undefined when x = 3 B. 0 C. 1 D. 4 4. What is lim f (x)? 1-3* A. 3 B. 1 C. 4 D. No limit exists as x approaches 3 from below. 5. Which of the following is a necessary condition for a function y = f(x) to be continuous at x = a? A. An asymptote MUST exist at x = a. B. The function must be defined at x = a. C. The limit of f(x) may does not exist at x = a. D. dy =0 atx = a. dx What is th of the following limit?D. dx -=0 atx = a. 6. What is the value of the following limit? (x7 -4) lim x-2 A. 2 B. co C. DNE D. 4 7. Let lim f (x) = Land limg (x) = M; which of the following is correct for lim (f (x) +8(x))? A. LIM B. MFL C. L. M D. L M 8. Which of the following represents the definition of the derivative of the function y =f(x)? A. dy lim f ( x-h) -f(x) dx h B. dy Jim f (xth)-f(x) dx C. = lim f ( xth)+f(x) dx h D. dy = lim f (x-h)+f(x) h 9. Let f (x) = -; what is the derivative of f(x)? A. x 2 C. 2Vx D. 2Vx10. How many turning points does the function y = (x). shown in the following graph have? A. 1 B. 2 C. 4 D. 3 11. What is the value of -- at any stationary point? dx A. DNE B. 0 C. Always negative D. Always positive 12. If y = f(x) and =0 at (x,.),) where (x. y, ) is a coordinate on f(x) then this dx coordinate is called A. An inflexion point B. A stationary point C. A point of discontinuity D. A derivative point 13. If - >0 at x= a, then f (x) dx A. Is increasing at x = a. B. Is decreasing at x = a. C. Is stationary at x = a. D. Has a vertical asymptote at x = a. 14. Which of the following is a step when determining the stationary point of the function y = f(x)? A. Determining the limit of f(x). B. Prove that the function is continuous. C. Determine the first derivative of f (x) D. Prove that the tangent line is sloping at the stationary point(s).[QUESTION 1] Consider the graph of y = f (x) shown below. Evaluate each limit or write DNE if the limit does not exist. [No justifications necessary for this problem]. h(x ) 10 10 12 14 A. lim / (x) = [2 marks] B. lim / (x) = [2 marks] C. lim f (x) = [2 marks] D. lim f (x) = [2 marks] E. lim f (x) = [2 marks] F. lim f (x) = [2 marks] G. For f (x) to be continuous at x = -4, we must set f (4)= [2 marks] H. Evaluate the limit of the following functions: (i) lim (3x# +7x-12) [2 marks] (ii) lim x - 2x -24 [4 marks] x-6 [Total: 20 Marks]2:02 4 X INCOURSE PDF - 253 KB C. lim f (x) = [2 marks] D. lim f (x) = [2 marks] E. lim f (x) = [2 marks] F. lim f (x) = [2 marks] G. For f (x) to be continuous at x = -4, we must set f ( 4) = [2 marks] H. Evaluate the limit of the following functions; lim (3x +7x-12) [2 marks] (ii) lim x - 2x -24 [4 marks] x-6 [Total: 20 Marks] [Question 2] A. Let f(x) = 2x' +x. Use the definition of the derivative to find f'(x). [10 marks] B. Find the derivative of each of the following functions and evaluate at the value of x indicated: i. y= or' +4x'-r'-x,x= 15 [4 marks] ii. y=7-x-x -2,x=3. [4 marks] iii. Using the sign of the derivative, tell if the graph of function in part (i) is increasing. decreasing or neither increasing nor decreasing. [2 marks] [Total: 20 marks] *#*END OF TEST*#*