Use your graphing calculator to graph each of the following functions. Use your mathematical skills to determine the real characteristics of the function. Does your calculator show these features? Include detailed sketches. 1. y = 3x 2 3 x 2
2. y = 0.075
FUNCTIONS (1.1)
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1. As you travel from Tucson to Bisbee (94 miles), you pass through Benson. Benson is 40 miles from Tucson. You can assume that you travel at a fairly constant speed. Sketch graphs to represent the functions below. Label the axes and
Holly, who is fortunate enough to live near the beach, takes a jog each evening. She takes a straight path between her house and the beach and back. The graph below shows her distance from home t minutes after 6:30 p.m. on one particular evening. Use the
READING GRAPHS OF FUNCTIONS 1. Use the graph below to answer the following: A. Find f (0) . B. Find f (7) . C. Find f (2) . D. Is f (6) positive or negative? E. Is f ( 1 2 ) positive or negative? F. Is f (1) > f (6) ? G. For what values of x is f ( x) = 0
Find an example of a function (in table, graph, or equation form) from a newspaper, magazine, or the internet. Cut it out or print it and answer the following questions on a separate piece of paper. Be sure to include your source. A. How do you know that
INDY 500 RACEWAY PROBLEM
Start
A racecar completes one lap at the Indy 500 Raceway. Assume the car starts at rest.
1. Sketch a graph of the cars total distance traveled as a function of time.
2. Sketch a graph of the cars distance to the starting line (as
FARENHEIT vs. CELSIUS A bank in town provides a sign with the current temperature. The temperature flashes alternately in Farenheit and Celsius. Degrees Degrees Celsius Farenheit 5 41 10 50 20 68 25 77 30 86
1. If we consider Farenheit as a function of Ce
LINEAR FUNCTIONS 1. The relationship between the tuition, T (in dollars), and the number of credits, c, at a particular college is given by 0c6 100 + 120c T (c ) = 800 + 120(c 6) 6 < c 18 A. What is the tuition for 6 credits? B. If the tuition was $1880,
EXPONENTIAL FUNCTIONS (1.2)
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1. Determine which table illustrates an exponential function and which one illustrates a linear function. Find formulas for these two functions, then find a formula for the third function.
x
-2 0 2 4 6
f ( x)
-25.22 3.50
INVERTIBLE FUNCTIONS 1. In each case, explain or verify that the given function is invertible. Find the inverse function.
A.
m f ( m)
1 2 3 4 5 0.09 2.10 5.60 7.80 9.40
B.
S (t ) = At 3 + K where A and K are constants.
C.
5
G(x)
0 0 10
x
2. The life expec
Kuta Software - Infinite Calculus
Name_
Newton's Method
Date_ Period_
Use two iterations of Newton's Method to approximate the real zeros of each function. Use the provided
initial guess.
2) y = x 5 2 x 3 + x 4
Guess: 1.8
1) y = cos 3 x 3 x
Guess: 0.4
y
y
Kuta Software - Infinite Calculus
Name_
Differentials
Date_ Period_
For each problem, find the differential dy.
1) y = x 3 2
2) y =
3
x
For each problem, find the general formulas for dy and y.
3) y = x 3 2
4) y =
2
x
For each problem, find dy and y, giv
Kuta Software - Infinite Calculus
Name_
Derivatives of Inverse Functions
For each problem, find ( f
)' ( x) by direct computation.
1
1) f ( x) = 3 x + 3
For each problem, find ( f
Date_ Period_
2) f ( x) = 2 x + 3
)' ( x) by using the theorem ( f 1)' ( x)
Kuta Software - Infinite Calculus
Name_
Implicit Differentiation
Date_ Period_
For each problem, use implicit differentiation to find
dy
in terms of x and y.
dx
1) 2 x 3 = 2 y 2 + 5
2) 3 x 2 + 3 y 2 = 2
3) 5 y 2 = 2 x 3 5 y
4) 4 x 2 = 2 y 3 + 4 y
5) 5 x 3
Kuta Software - Infinite Calculus
Name_
Differentiation - Logs and Exponentials
Date_ Period_
Differentiate each function with respect to x.
1) y = 4
4x4
2) y = 4
3) y = log 3 3 x 2
5) y = log 3 (3 x 5 + 5)
7) y = (4 + 2)
x3
9) y = 3
5 x 3
4) y = log 2 4
Kuta Software - Infinite Calculus
Name_
Differentiation - Trigonometric Functions
Date_ Period_
Differentiate each function with respect to x.
1) f ( x) = sin 2 x 3
2) y = tan 5 x 3
3) y = sec 4 x 5
4) y = csc 5 x 5
5) y = (2 x 5 + 3)cos x 2
2 x 2 5
6) y
Kuta Software - Infinite Calculus
Name_
Differentiation Rules, with Tables
Date_ Period_
For each problem, you are given a table containing some values of differentiable functions f ( x), g( x) and
their derivatives. Use the table data and the rules of di
Kuta Software - Infinite Calculus
Name_
Differentiation - Chain Rule
Date_ Period_
Differentiate each function with respect to x.
1) y = ( x 3 + 3)
5
3) y = (5 x 3 3)
2) y = (3 x 5 + 1)
3
4) y = (5 x 2 + 3)
4
5) f ( x) =
4
3 x 4 2
6) f ( x) =
7) f ( x) =
Kuta Software - Infinite Calculus
Name_
Differentiation - Product Rule
Date_ Period_
Differentiate each function with respect to x.
1) y = x 3 (3 x 4 2)
2) f ( x) = x 2 (3 x 2 2)
3) y = (2 x 4 3)(2 x 2 + 1)
4) f ( x) = (2 x 4 3)( x 2 + 1)
5) f ( x) = (5 x
Kuta Software - Infinite Calculus
Name_
Higher Order Derivatives
Date_ Period_
For each problem, find the indicated derivative with respect to x.
1) y = x 2
Find
d 2y
dx 2
2) f ( x) = 4 x 3
Find f '
3) y = 4 x
d 3y
Find 3
dx
4) f ( x) = 5 x 4
Find f '
5)