Transcript Document 7571572

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)
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PreCalc Review
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Unit A - PreCalculus Review
Santowski - Calculus
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Fast Five
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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 log3324 - log34
• 8. Expand using LoL
4
a
log 2
bc
• 9. Evaluate log79 + log35 using GDC
• 10. Solve 3x = 11

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Fast Five
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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∞ (0.333333333-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?
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Fast Five
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1. Simplify and state restrictions: 4t 2  8t
2t
2x 2  x 15
0
x 2
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3. Solve
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4. Find the VA and HA for
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 Draw any graph that is discontinuous at x = 2
5.
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6. Find g(2) if g(x) 
5  2x
4x  6

x 2
 x 2  4
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7. State the restrictions on x for
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9. Evaluate
g(2.0001) if g(x) 
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10. Evaluate g(1,000,000)
 if
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
2x
x 2  8x  15
x 2
x2  4
x 2 1
g(x) 
3  2x 2
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Fast Five
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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)2 + 8. 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
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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)
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10. Evaluate |-5| + |4| - 2|-3|
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Fast Five
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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
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Fast Five
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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?
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(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
1
2
• (4) Express ln x  4 ln y  ln x 1
2
as a single log
• (5) Solve log2x + log4x + log8x = 11

• (6) Express
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 x 
ln

 x  1 
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as a sum/diff of logs
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(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)
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(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.
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(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?
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(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?
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(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
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(A) Algebra
• Solve 2x3 – 9x2 - 8x = -15 algebraically
• Solve 2x3 + 14x - 20 = 9x2 - 5 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
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(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.
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(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
y  x2
y  6 x
y  2x 2  5x 12
1
y
x3
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(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
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(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)
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(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
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(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.
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(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.
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(A) Algebra
• Given the following rational equation,
simply and
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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
6x 13x  6
f (x) 
2
8x  6x  9
2
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(A) Algebra
• Solve the following QE using the any of the three
algebraic methods:
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(a) 2x2 + 7x = 4
(b) 2x2 - 6x + 4 = 0
(c) 2x2 = -1 + 6x
(d) 2x2 + 7x = 4
(e) 2x2 - 6x + 4 = 0
(f) 2x2 = -1 + 6x
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(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
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(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 + 100 (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
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(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?
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(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 slopeintercept form.
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(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
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(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)
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(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?
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(B) Graphs
• Be able to identify asymptotes, intercepts, end
behaviour, domain, range for y = ax
• Ex. Given the function y = 2 + 3-x, determine the
following:
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- domain and range
- asymptotes
- intercepts
- end behaviour
- sketch and then state intervals of increase/decrease as
well as concavities
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(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 – 900000 = 0,
explain to your neighbour how to graph this line (i)
using a TI-89 GDC and then (ii) graphing by hand
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(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:
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- domain and range
- asymptotes
- intercepts
- end behaviour
- sketch and then state intervals of increase/decrease as
well as concavities
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(C) Calculus Concepts
• (1) Explain what the equation for f(x) means
1 x 2 , x  2
f (x)  
2x  5, x  2
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(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?
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(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.
lim 2  ex 
x 
lim  e
x
tan x


2
3h 1
lim
h 0
h
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(C) Calculus Concepts
• We can also introduce limits that correspond to “end behaviour”
• So given the expression
6x 2 13x  6
lim
:
x  8x 2  6x  9
• (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
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6x 2 13x  6
lim
x  8x 2  6x  9
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(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.
lim ln x  4 
x 
lim  ln x  4 
x 4
lim  ln sin x 
x 
lim log 10 x  x 
2
x 1
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(C) Calculus Concepts
• Ex 3. Given the function :
ln x
f (x) 
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
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?
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(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
f (x)  ln x 1
2
• From the graph, determine: domain, range, max
and/or min, where f(x) is increasing, decreasing,
concave up/down, asymptotes

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(C) Calculus Concepts
2x 2  8
f (x)  2
x  x 2
• For
TI-89 and determine:
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generate its graph on the
(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?
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(C) Calculus Concepts
• So given the expression
6x 2 13x  6
:
lim
2
3
x  8x  6x  9
2
• (B1) Direct Substitution: Evaluate g(-1.5) and explain
• (B2) Substitution: Evaluate g(1.5000001) and g(1.499999) 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
6x 2 13x  6
lim
2
3
x  8x  6x  9
4
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(C) Calculus Concepts
• Ex 4. Given the function f(x) = x2e-x, find the average rate of
change of f(x) between:
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(a) 1 and 1.5
(b) 1.4 and 1.5
(c) 1.499 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?
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(C) Calculus Concepts
• A function is defined as follows:
 e x  a
x  2

f (x)   x  2 2  x  3
b  e x3
x3

• (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
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(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
x 2
f (x)  e
• From the graph, determine: domain, range, max
and/or min, where f(x) is increasing, decreasing,
concave up/down, asymptotes

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(C) Calculus Concepts
• (1) Define a piecewise function as f(x) where
 x  c, x  3

f (x)  2  bx, 3  x  1
x 3  bx, x  1


• (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
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(C) Calculus Concepts
•
ln x
Ex 4. Given the function , f (x) 
x
of f(x) between:
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find the average rate of change
(a) 1 and 1.5
(b) 1.4 and 1.5
 1.5
(c) 1.499 and
(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?
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(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?
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(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)
•
•
•
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QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
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