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MCV4U V.1 UNIT 1 TEST: INTRODUCTION TO CALCULUS Name: COMMUNICATION /6 + /4 Date: COMMUNCATION - TEST /12 APPLICATION THINKING /12 /11 (2 MARKS) Enough steps shown to clearly demonstrate thinking Solutions that are neat and easy to follow Proper use of mathematical symbols and notation Equal signs (aligned, one per line) Side calculations clearly labeled if used Units used as required Variables and functions defined as required COMMUNCATION - UPLOAD KNOWLEDGE Concluding statements used as required Fractions, not decimals, unless otherwise stated Radicals, not decimals, unless otherwise stated Fractions reduced to lowest terms Denominators rationalized Labels, arrows, smooth lines, and ruler used as required Labelled axis and titled graph (2 MARKS) Units used as required All work submitted as one PDF file (-1 mark) PDF file follows proper naming convention LastName_FirstName_Chapter#_Test/Quiz PDF files must be in portrait mode PDF file must be submitted in the appropriate assignment / quiz location. (-2 marks) PDF must contain the full page PDF is in order (-1 mark) PDF must be clear and readable (-2 marks) UNSUPPORTED WORK WILL RECEIVE A MARK OF ZERO FOR THAT QUESTION. COMMUNICATION: 1. Explain the following statement: A function does not need to be continuous at the point x=a in order for the limit to exist at the point x=a. Provide an equation and a sketch of such a function. (4) 2. The limit below represents the derivative of some function f(x) evaluated at some point x a Determine the function f(x) and the value of a. 2(6 + ℎ)2 − 𝑠(6)2 lim ℎ→0 ℎ (2) KNOWLEDGE: 3. Use the graph to estimate the limits and value of the function, or explain why the limits do not exist. lim 𝑓(𝑥) 𝑥→2+ lim 𝑓(𝑥) lim 𝑓(𝑥) 𝑥→2− lim 𝑓(𝑥) 𝑥→1− 𝑥→3 lim 𝑓(𝑥) 𝑓(2) 𝑥→2 (4) State any points of discontinuity. 4. Evaluate each of the following limits. a) 𝑥 2 −𝑎2 , 𝑥→𝑎 √𝑥−√𝑎 lim 𝑎>0 b) lim ((4𝑥 − 𝑥→3 2 ) (6 + 𝑥−3 𝑥 − 𝑥 2 )) (4) 5. Find the equation of the tangent line to the curve 𝑦 = √𝑥 + 1 at 𝑥 = 3. Graph the function and the tangent line. Label the function and tangent line on the graph. No decimals. (4) APPLICATION: 6. Let 𝑥 , 𝑖𝑓 𝑥 < 1 3 , 𝑖𝑓 𝑥 = 1 𝑔(𝑥) = 2 − 𝑥 2 , 𝑖𝑓 1 < 𝑥 ≤ 2 3, , 𝑖𝑓 𝑥 > 2 { Evaluate each limit, if it exits. a) lim− 𝑔(𝑥) 𝑥→1 (3) b) lim 𝑔(𝑥) 𝑥→1 c) lim 𝑔(𝑥) 𝑥→1+ 7. For what values of A and B is this statement correct? Justify your response. √𝐴𝑥 + 𝐵 − 3 lim =1 𝑥→0 𝑥 (2) 8. A carpenter is constructing a large cubical storage shed. The volume of the shed is given by 16−𝑥 2 𝑉(𝑥) = 𝑥 2 ( ), where x is the side length in metres. Determine the instantaneous rate of 4𝑥 change of volume when the side length is 2 metres. (3) 9. Sketch the graph of a function g x , such that the following characteristics are satisfied: lim 𝑔(𝑥) = −2 𝑥→2− lim 𝑔(𝑥) = 0 𝑥→−2+ lim 𝑔(𝑥) = 3 𝑥→0+ lim 𝑔(𝑥) = +∞ 𝑥→3+ lim 𝑔(𝑥) = 0 𝑥→−∞ (4) 𝑔(−2) = 0 lim 𝑔(𝑥) = 3 𝑥→0− lim 𝑔(𝑥) = +∞ 𝑥→3− lim 𝑔(𝑥) = +∞ 𝑥→+∞ THINKING: 10. For what values of c is the function 𝑔(𝑥) continuous at every number? (𝑐𝑥 − 1)3 , 𝑔(𝑥) = { 2 2 𝑐 𝑥 − 1, −∞ < 𝑥 < 2 2≤𝑥<∞ (3) 11. Sketch the function of 𝑓(𝑥), and determine lim 𝑓(𝑥). 𝑥→1 Show all you work. 2 + 𝑥, 𝑓(𝑥) = {1, 𝑥 − 1, 𝑥<1 𝑥=1 𝑥>1 (4) 12. Determine the 4 tiered piecewise equation of the following function. (Question 11 is a 3 tiered piecewise function). (4)