1. PreCALC Unit Objectives
1. Review characteristics (algebraic & graphic) of fundamental functions (R)
2. Review/Extend application of function models (R/E)
3. Introduce new function concepts pertinent to Calculus (N)
9/21/2018
PreCalc Review

2. Unit A - PreCalculus Review
Santowski - Calculus
9/21/2018
PreCalc Review

3. Fast Five 1. Solve ln(2x - 5) = -3 2. Sketch f(x) = -ln(2 - x)
3. State the domain of f(x) = (ln(2 - x))0.5
4. Evaluate log232 + log31/81+ log0.2564
5. Sketch the inverse of f(x) = 3 - log2x
6. Find the domain of log3(32 - 2x2)
7. Evaluate log log34
8. Expand using LoL
9. Evaluate log79 + log35 using GDC
10. Solve 3x = 11
9/21/2018
PreCalc Review

4. Fast Five 1 Solve 2-x+2 = 0.0625 2. Sketch a graph of y = (0.5)x + 3
3. Solve 4x2 - 4x - 15 = 0
4. Evaluate limx∞ ( x)
5. Solve 3x+2 - 3x = 216
6. Solve log4(1/256) = x
7. Evaluate limx3 ln(x - 3)
8. Solve 22x + 2x - 6 = 0
9. State the exact solution for 2x-1 = 5 (2 possible answers)
10. Is f(x) = 3 + e-x an increasing or decreasing function?
9/21/2018
PreCalc Review

5. Fast Five 1. Simplify and state restrictions: 3. Solve
4. Find the VA and HA for
5. Draw any graph that is discontinuous at x = 2
6. Find g(2) if
7. State the restrictions on x for
9. Evaluate g(2.0001) if
10. Evaluate g(1,000,000) if
9/21/2018
PreCalc Review

6. Fast Five
1. Sketch a graph that increases on (-2,4) U (5,8) and decreases on (4,5)
2. Solve 2x2 - 12x - 7 = 0
3. Sketch the graph of a decreasing exponential function
4. How many roots does Q(x) = 3x2 + 4x - 8 have?
5. Evaluate tan(π/4)
6. Find the roots and vertex of f(x) = -0.5(x - 2) Sketch a graph that is positive and increasing at x = 2
7. Sketch the graph of f(x) from Q6.
8. Factor 6x2 + x - 15
9. At what values is f(x) = (x-4)/(x-3) equal to 0?
10. Simplify x/5 + 5/x
9/21/2018
PreCalc Review

7. Fast Five 1. Evaluate f(3) if f(x) = x3 + x - 3
2. Solve 0 = (x + 3)(2x2 - 16)
3. Sketch a function that is concave up on [2,8], as well as increasing on (2,6) and decreasing on (6,8)
4. Sketch a quartic polynomial which has a negative leading coefficient
5. Find the vertical and horizontal asymptotes of
f(x) = 1/(x + 2) - 4
6. Evaluate log327 ¸ 2-3
7. State the domain and range of y = -2(x + 1)2 + 3
8. Is x + 2 a factor of x3 - x2 +2x - 1?
9. Solve 0 < (2x + 1)(4 - x)
10. Evaluate |-5| + |4| - 2|-3|
9/21/2018
PreCalc Review

8. Fast Five 1. Determine the slope of the line 3x - 4y = 12
2. Determine the slope of the line passing through the points (2,4) and (-1/2, -5)
3. Find k so that the line kx + 3y = 6 is perpendicular to a line having a slope of 4
4. The equation of a line through the point (1,-3) having a slope of 4 is …
5. Sketch a graph of v(t) = 5 - 2t
6. Find f-1(x) if f(x) = 0.25x + 3
7. The x-intercepts of g(x) = x2 - 2x - 8 are …
8. Write the equations of any two lines that are parallel
9. Write the equations of any two lines that are perpendicular
10. Determine the slope, x- & y-intercepts of the line x/2 - y/5 = 1
9/21/2018
PreCalc Review

9. Fast Five 1. Name the type of function: f(x) = x3
2. Find f(2) for f(x) = x3
3. Name the type of function: g(x) = 3x
4. Find g(2) for g(x) = 3x
5. Sketch the graph of h(x) = x2
6. Find h-1(2) for h(x) = x2
7. At what values is t(x) = (x - 4)/(x - 3) undefined
8. Sketch a graph of a linear function with a positive y-intercept and a negative slope
9. Evaluate sin(/2) - cos(/3)
10. Sixty is 30% of what number?
9/21/2018
PreCalc Review

10. (A) Algebra (1) Find the inverse of f(t) = 67.38(1.026)t
(2) Solve e3-2x = 4
(3) Solve log6x + log6(x - 5) = 2
(4) Express as a single log
(5) Solve log2x + log4x + log8x = 11
(6) Express as a sum/diff of logs
9/21/2018
PreCalc Review

11. (A) Algebra
Ex 1. Determine the equation of a line parallel to 3x – 6y + 11 = 0 passing through the point (-1,3)
Ex 2. Determine the equation of a line perpendicular to 3x – 6y + 11 = 0 passing through the point (-1,3)
9/21/2018
PreCalc Review

12. (A) Algebra
Mario owns a construction company and occasionally needs to rent a small front end loader. ACE rentals charges $75 per day for the rental plus an initial insurance fee of $50.
(a) Determine the slope and the y-intercept and interpret their meanings.
(b)Write the word equation and the algebraic expression that represents this relationship
(c) Make a table of values
(d) Draw the corresponding graph.
9/21/2018
PreCalc Review

13. (A) Algebra
Find k so that when x2 + 8x + k is divided by x - 2, the remainder is 3
Find the value of k so that when x3 + 5x2 + 6x + 11 is divided by x + k, the remainder is 3
When P(x) = ax3 – x2 - x + b is divided by x - 1, the remainder is 6. When P(x) is divided by x + 2, the remainder is 9. What are the values of a and b?
9/21/2018
PreCalc Review

14. (A) Algebra ex 1. Show that x - 2 is a factor of x3 - 7x + 6
ex. 2. Show that -2 is a root of 2x3 + x2 - 2x + 8 = 0. Find the other roots of the equation. (Show with GC)
ex. 4. Is x - √2 a factor of x4 – 5x2 + 6?
9/21/2018
PreCalc Review

15. (A) Algebra Examples: Factor 2x3 + 13x2 + 5x - 6
Factor 6x4 - 19x3 - 2x2 + 44x - 24
Now use the TI-89 to verify your factors (i) algebraically (ii) numerically (iii) graphically
9/21/2018
PreCalc Review

16. (A) Algebra Solve 2x3 – 9x2 - 8x = -15 algebraically
Solve 2x4 - 3x3 + 2x2 - 6x - 4 = 0 then graph using roots, points, end behaviour. Approximate turning points, max/min points, and intervals of increase and decrease.
Of course, we can also simply use the TI-89 and solve these (ii) algebraically, (ii) numerically, (iii) graphically
9/21/2018
PreCalc Review

17. (A) Algebra
The owner of an apartment building is going to buy a new air conditioner for the building. One brand costs $3000 to buy and $40 per month to operate. Another model costs $3600 to buy and $25 per month to operate. Determine which AC is the owner should buy. Explain your choice.
9/21/2018
PreCalc Review

18. (A) Algebra
Find the domain and range of the following functions. Work without the GDC to start with and then verify/check your answers using the GDC
9/21/2018
PreCalc Review

19. (A) Algebra
Mr. Santowski tutors students to make some extra money. He charges a fee of $50 for the first hour (or part thereof). For each addition hour (or part thereof), he charges $15. Let F(h) represent the fees charged to tutor Calculus students for h hours.
(a) Graph F(h) on the interval (0,6]
(b) State the domain and range
(c) Does it seem reasonable that F(h) is a function? What would be the consequence to you as a student if F(h) was NOT a function
9/21/2018
PreCalc Review

20. (A) Algebra (1) Factor e2x - ex
(2) Factor and solve xex - 2x = 0 algebraically. Give exact and approximate solutions (CF)
(3) Factor 22x - x2 (DOS)
(4) Express 32x - 5 in the form of a3bx (EL)
(5) Solve 3e2x - 7ex + 4 = 0 algebraically. Give exact and approximate solutions (F)
(6) Solve 4x + 5(2x) - 12 = 0 algebraically. Give exact and approximate solutions (F)
9/21/2018
PreCalc Review

21. (A) Algebra
(1) Give a graphic interpretation of the inequality 3 - 2x > x - 2
(2) Solve algebraically
(3) Verify algebra with a graphic solution
(4) Solve the inequality 4 - 3x > ax + 5
(5) When is the inequality 4 - 3x > ax + 5 not true. Explain why
9/21/2018
PreCalc Review

22. (A) Algebra
(1) Give a graphic interpretation of the inequality x2 - 8 > -2 + x
(2) Solve algebraically
(3) Verify algebra with a graphic solution
(4) Under what conditions for b is x2 - b2x + b2 > 0.
9/21/2018
PreCalc Review

23. (A) Algebra (1) Solve x3 - 3x2 -10x + 24 > 0 algebraically
(2) Verify your solution 3 different ways on the TI-89.
(3) An open topped box is made by cutting out squares of x cm from the corners of a paper that originally measured 24 cm by 10 cm. What are the dimensions of the box such that its volume is at least 100 cm3.
9/21/2018
PreCalc Review

24. (A) Algebra
Given the following rational equation, simply and state the restrictions on x
Solve f(x) = 0 and state the graphic significance of your solution
Find the we will focus primarily on vertical and horizontal asymptotes
9/21/2018
PreCalc Review

25. (A) Algebra
Solve the following QE using the any of the three algebraic methods:
(a) 2x2 + 7x = 4
(b) 2x2 - 6x + 4 = 0
(c) 2x2 = x
(d) 2x2 + 7x = 4
(e) 2x2 - 6x + 4 = 0
(f) 2x2 = x
9/21/2018
PreCalc Review

26. (A) Algebra
To find the max/min of a parabola, we have three algebraic methods we can use:
(1) completing the square method
(2) averaging the roots to find the axis of symmetry and then evaluating the function at that x-value
(3) finding the axis of symmetry and then evaluating the function at that x-value
Example:
Find the maximum point of f(x) = -3x2 - 12x - 9
9/21/2018
PreCalc Review

27. (A) Algebra
A shoe store owner has found that her revenue from selling x shoes (where x is in thousands and R(x) is in hundreds of thousands) is modeled by the equation R(x) = -x2 + 30x, while the cost of producing the shoes in her factory is modeled by the equation C(x) = 5x  (where x is in thousands and C(x) is in tens of thousands)
(a) Find the minimum break-even quantity
(b) Find the maximum revenue
(c) Find the maximum profit
9/21/2018
PreCalc Review

28. (A) Algebra
I own an apartment building which has 1300 units, for which I charge $700 rent per month. At this rent, all units are occupied. I know that for every $20 rent increase, I will lose 25 tenants. What rent will maximize my revenue? What will be the maximum revenue? How many units are now vacated?
9/21/2018
PreCalc Review

29. (A) Algebra
Ex 1. Given the following information about lines, determine their equations, expressing the equations in slope-intercept form, in slope-point form and in general form. Include a diagram/sketch
(i) passing through (2,-5) having a slope of –¼.
(ii) passing through (-3,-4) and (-5,6)
(ii) passing through (a,b) having an angle of 30o with the positive x-axis
Ex 2. Write the equation Ax + By + C = 0 in slope-intercept form.
9/21/2018
PreCalc Review

30. (B) Algebra & Graphs
(1) Draw a polynomial of degree 3 or more such that P(x) is increasing on the domain of |x| > 4
(2) Draw a polynomial of degree 3 or more such that P(x) is concave down on the domain of |x + 2| < 1
(3) Draw a function that is concave up and increasing and is only defined on the domain of |x - 3| > 5
9/21/2018
PreCalc Review

31. (B) Graphs Sketch a graph of a graph that has the following features:
(i) f is a function whose rate of change increases on the interval (-4,6)
(ii) This function, f , has its rate of change decreasing on (6,+∞ )
(iii) it goes through the point (2,-1)
9/21/2018
PreCalc Review

32. (B) Graphs
Next, we want to investigate polynomials and find their intervals of increase/decrease/concavity and end behaviour
Use the TI-89 to graph P(x) = -2x4 + 5x3 + 4x2 - 3x in an appropriate view window.
Then find where P(x) increases and decreases. What key points are you looking for?
9/21/2018
PreCalc Review

33. (B) Graphs
Be able to identify asymptotes, intercepts, end behaviour, domain, range for y = ax
Ex. Given the function y = x, determine the following:
- domain and range
- asymptotes
- intercepts
- end behaviour
- sketch and then state intervals of increase/decrease as well as concavities
9/21/2018
PreCalc Review

34. (B) Graphs
Given the linear equation 2x – 5y –12 = 0, explain to your neighbour how to graph this line (i) using a TI-89 GDC and then (ii) graphing by hand
Given the linear equation 150x + 300y – = 0, explain to your neighbour how to graph this line (i) using a TI-89 GDC and then (ii) graphing by hand
9/21/2018
PreCalc Review

35. (B) Graphs
Be able to identify asymptotes, intercepts, end behaviour, domain, range for y = logax
Ex. Given the function y = log2(x - 1) - 2, determine the following:
- domain and range
- asymptotes
- intercepts
- end behaviour
- sketch and then state intervals of increase/decrease as well as concavities
9/21/2018
PreCalc Review

36. (C) Calculus Concepts (1) Explain what the equation for f(x) means
(2) Use the TI-89 to graph f(x)
(3) Solve f(x) > 0
(4) Where is f(x) concave up?
(5) Where is increasing?
(6) Evaluate f(1.9) and evaluate f(2.1)
(7) Is f(x) a continuous function?
9/21/2018
PreCalc Review

37. (C) Calculus Concepts
Ex 2. Evaluate the following limits numerically or algebraically. Interpret the meaning of the limit value. Then verify your limits and interpretations graphically.
9/21/2018
PreCalc Review

38. (C) Calculus Concepts
We can also introduce limits that correspond to “end behaviour”
So given the expression :
(B1) Direct Substitution: Is it possible?? Explain
(B2) Substitution: Evaluate g(-20,000) and g(-2,000,000) and explain
(B3) Simplify algebraically and again evaluate g(-2,000,000)
(B4) Graphic: Graph the function and zoom out
Now repeat for
9/21/2018
PreCalc Review

39. (C) Calculus Concepts
Ex 2. Evaluate the following limits numerically or algebraically. Interpret the meaning of the limit value. Then verify your limits and interpretations graphically.
9/21/2018
PreCalc Review

40. (C) Calculus Concepts Ex 3. Given the function :
(i) find the intervals of increase/decrease of f(x)
(ii) is the rate of change at x = 2 equal to/more/less than the rate at x = 1? How do you know?
(iii) find intervals of x in which the rate of change of the function is increasing. Explain why you are sure of your answer.
(iv) where is the rate of change of f(x) equal to 0? Explain how you know that?
9/21/2018
PreCalc Review

41. (C) Calculus Concepts
Now we will apply the concepts of limits, continuities, rates of change, intervals of increase/decreasing & concavity to exponential function
Ex 1. Graph
From the graph, determine: domain, range, max and/or min, where f(x) is increasing, decreasing, concave up/down, asymptotes
9/21/2018
PreCalc Review

42. (C) Calculus Concepts
For generate its graph on the TI-89 and determine:
(1) factor either numerator/denominator
(2) intervals of increase & decrease
(3) intervals of concavity
(4) zeroes/roots
(5) equations of vertical and horizontal asymptotes
(6) is the function continuous or discontinuous?
9/21/2018
PreCalc Review

43. (C) Calculus Concepts So given the expression :
(B1) Direct Substitution: Evaluate g(-1.5) and explain
(B2) Substitution: Evaluate g( ) and g( ) and explain
(B3) Simplify algebraically and again evaluate g(1.5)
(B4) Graphic: Graph the function and zoom in at x = 1.5
Now repeat for
9/21/2018
PreCalc Review

44. (C) Calculus Concepts
Ex 4. Given the function f(x) = x2e-x, find the average rate of change of f(x) between:
(a) 1 and 1.5
(b) 1.4 and 1.5
(c) and 1.5
(d) predict the rate of change of the fcn at x = 1.5
(e) evaluate limx1.5 x2e-x.
(f) Explain what is happening in the function at x = 1.5
(g) evaluate f(1.5)
(h) is the function continuous at x = 1.5?
9/21/2018
PreCalc Review

45. (C) Calculus Concepts A function is defined as follows:
(i) Evaluate limx-2 if a = 1
(ii) Evaluate limx3 if b = 1
(iii) find values for a and b such f(x) is continuous at both x = -2 and x = 3
9/21/2018
PreCalc Review

46. (C) Calculus Concepts
Now we will apply the concepts of limits, continuities, rates of change, intervals of increase/decreasing & concavity to exponential function
Ex 1. Graph
From the graph, determine: domain, range, max and/or min, where f(x) is increasing, decreasing, concave up/down, asymptotes
9/21/2018
PreCalc Review

47. (C) Calculus Concepts (1) Define a piecewise function as f(x) where
(a) Find values for b and c such that f(x) is continuous at -3 but not at 1
(b) Find value(s) for b such that f(x) is continuous at 1 but not at -3
(c) Find values for b and c such that f(x) is continuous on x€R
9/21/2018
PreCalc Review

48. (C) Calculus Concepts
Ex 4. Given the function , find the average rate of change of f(x) between:
(a) 1 and 1.5
(b) 1.4 and 1.5
(c) and 1.5
(d) predict the rate of change of the fcn at x = 1.5
(e) evaluate limx1.5 f(x).
(f) Explain what is happening in the function at x = 1.5
(g) evaluate f(1.5)
(h) is the function continuous at x = 1.5?
(i) is the function continuous at x = 0?
9/21/2018
PreCalc Review

49. (C) Calculus Concepts Ex 3. Given the function f(x) = x2e-x:
(i) find the intervals of increase/decrease of f(x)
(ii) is the rate of change at x = -2 equal to/more/less than the rate of change equal to/greater/less than the rate at x = -1?
(iii) find intervals of x in which the rate of change of the function is increasing. Explain why you are sure of your answer.
(iv) where is the rate of change of f(x) equal to 0? Explain how you know that?
9/21/2018
PreCalc Review

50. (C) Calculus Concepts
Ex 1: Given the parabolas on the right, state the intervals of increase and decrease
Ex 2: Given the equations of the following parabolas, determine the intervals of increase and decrease
F(x) = -(x + 3)2 + 1
G(x) = 2x2 + x - 10
H(x) = (x + 4)(2 - 3x)
9/21/2018
PreCalc Review