Lesson 1 – A.1.1 – Function Characteristics Calculus - Santowski 9/4/2015 Calculus - Santowski 1.

Lesson 1 – A.1.1 – Function Characteristics

Calculus - Santowski04/19/23Calculus - Santowski

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Lesson Objectives

•1. Review characteristics of functions - like domain, range, max, min, intercepts

•2. Extend application of function models•3. Introduce new function concepts

pertinent to Calculus - concepts like intervals of increase, decrease, concavity, end behaviour, rate of change

04/19/23Calculus - Santowski

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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?


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(A) Function Characteristics•Terminology to review:

•Domain•Range•Symmetry•Roots, zeroes•Turning point•Maximum, minimum•Increase, decrease•End behaviour


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(A) Function Characteristics• Domain: the set of all possible x values

(independent variable) in a function

• Range: the set of all possible function values (dependent variable, or y values)

• to evaluate a function: substituting in a value for the variable and then determining a function value. Ex f(3)

• finite differences: subtracting consecutive y values or subsequent y differences


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(A) Function Characteristics•zeroes, roots, x-intercepts: where the

function crosses the x axes (y-value is 0)

•y-intercepts: where the function crosses the y axes (x-value is 0)

•direction of opening: in a quadratic, curve opens up or down

•symmetry: whether the graph of the function has "halves" which are mirror images of each other


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(A) Function Characteristics•turning point: points where the direction

of the function changes

•maximum: the highest point on a function

•minimum: the lowest point on a function

•local vs absolute: a max can be a highest point in the entire domain (absolute) or only over a specified region within the domain (local). Likewise for a minimum.


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(A) Function Characteristics• increase: the part of the domain (the interval)

where the function values are getting larger as the independent variable gets higher; if f(x1) < f(x2) when x1 < x2; the graph of the function is going up to the right (or down to the left)

• decrease: the part of the domain (the interval) where the function values are getting smaller as the independent variable gets higher; if f(x1) > f(x2) when x1 < x2; the graph of the function is going up to the left (or down to the right)

• "end behaviour": describing the function values (or appearance of the graph) as x values getting infinitely large positively or infinitely large negatively


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(B) Working with the Function Characteristics

•The slides that follow simply review all functions that you have seen to date in previous courses

•You are expected to become proficient with a method of GRAPHICALLY determining that which is being asked of you (Use TI-89)

•Work with your partners through the following exercises:


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(B) Working with the Function Characteristics• For the quadratic functions, determine the

following:• f(x) = -½x² - 3x - 4.5 f(x) = 2x² - x + 4

• (1) Leading coefficient (2) degree • (3) domain and range (4) evaluating f(-2) • (5) zeroes or roots (6) y-intercept• (7) Symmetry (8) turning points• (9) maximum values (local and absolute)• (10) minimum values (local and absolute)• (11) intervals of increase and intervals of

decrease• (12) end behaviour (+x) and end behaviour

(-x)


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(B) Working with the Function Characteristics• For the cubic functions, determine the following:• f(x) = x3 - 5x² + 3x + 4• f(x) =-2x3 + 8x² - 5x + 3• f(x) = -3x3-15x² - 9x + 27


decrease• (12) end behaviour (+x) and end behaviour (-x)


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(B) Working with the Function Characteristics• For the quartic functions, determine the

following:• f(x)= -2x4-4x3+3x²+6x+9• f(x)= x4-3x3+3x²+8x+5• f(x) = ½x4-2x3+x²+x+1




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• For the Root Functions, determine the:




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y = 6 − x3 and y = 2x 2 + 5x −12


• For the Rational Functions, determine the:




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y =1

x + 3 and y =

x 2 − 4

x + 2 and y =

−2

x 2 − x − 6


• For the Exponential Functions, determine the:




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y = ex and y = x 2ex and y =1

ex2


• For the logarithmic functions, determine the:




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y = ln(9 − x) and y = ln(1+ x 2) and y = ln(x 2 − x − 6)


• For the trig functions, determine the:




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y = sin(2x) and y = sin(2x)cos(x) and y = tan(x)

(C) “A” Level Function Questions

• 1. If and if f(2) = 2, find the value of f(1).

• 2. Suppose that

• Then let g1(x) = f(x) and g2(x) = f(f(x)) and so on such that gn(x) = f( ….. (f(x)) …) …. where f occurs n times here

• Develop a general formula for gn(x) and suggest a method for proving that your general formula is true for all cases of n.


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f (x +1) =2 f (x) +1

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f (x) =x

1− x

Lesson 1 – A.1.1 – Function Characteristics Calculus - Santowski 9/4/2015 Calculus - Santowski 1.

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