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# Lesson 2: A Catalog of Essential Functions (handout)

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We introduce a number of different functions that can be used for modeling.

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### Lesson 2: A Catalog of Essential Functions (handout)

1. 1. . V63.0121.001, Calculus I . Sec on 2.2 : Essen al Func ons . January 26, 2011 Professor Ma hew Leingang . Sec on 2.2 Notes A Catalogue of Essen al Func ons V63.0121.001, Calculus I Professor Ma hew Leingang New York University Announcements First WebAssign-ments are due January 31 First wri en assignment is due February 2 Do the Get-to-Know-You survey for extra credit!. . Announcements Notes First WebAssign-ments are due January 31 First wri en assignment is due February 2 Do the Get-to-Know-You survey for extra credit!. . Objectives Notes Iden fy diﬀerent classes of algebraic func ons, including polynomial (linear, quadra c, cubic, etc.), polynomialra onal, power, trigonometric, and exponen al func ons. Understand the eﬀect of algebraic transforma ons on the graph of a func on. Understand and compute the composi on of two func ons.. . . 1.
2. 2. . V63.0121.001, Calculus I . Sec on 2.2 : Essen al Func ons . January 26, 2011 Professor Ma hew Leingang What is a function? Notes Deﬁni on A func on f is a rela on which assigns to to every element x in a set D a single element f(x) in a set E. The set D is called the domain of f. The set E is called the target of f. The set { y | y = f(x) for some x } is called the range of f.. . Classes of Functions Notes linear func ons, deﬁned by slope an intercept, point and point, or point and slope. quadra c func ons, cubic func ons, power func ons, polynomials ra onal func ons trigonometric func ons exponen al/logarithmic func ons. . Outline Notes Algebraic Func ons Linear func ons Other Polynomial func ons Other power func ons General Ra onal func ons Transcendental Func ons Trigonometric Func ons Exponen al and Logarithmic func ons Transforma ons of Func ons Composi ons of Func ons. . . 2.
3. 3. . V63.0121.001, Calculus I . Sec on 2.2 : Essen al Func ons . January 26, 2011 Professor Ma hew Leingang Linear functions Notes Linear func ons have a constant rate of growth and are of the form f(x) = mx + b. Example In New York City taxis cost \$2.50 to get in and \$0.40 per 1/5 mile. Write the fare f(x) as a func on of distance x traveled. Answer If x is in miles and f(x) in dollars, f(x) = 2.5 + 2x. . Notes Example Biologists have no ced that the chirping rate of crickets of a certain species is related to temperature, and the rela onship appears to be very nearly linear. A cricket produces 113 chirps per minute at 70 ◦ F and 173 chirps per minute at 80 ◦ F. (a) Write a linear equa on that models the temperature T as a func on of the number of chirps per minute N. (b) If the crickets are chirping at 150 chirps per minute, es mate the temperature.. . Solution Notes Solu on. . . 3.
4. 4. . V63.0121.001, Calculus I . Sec on 2.2 : Essen al Func ons . January 26, 2011 Professor Ma hew Leingang Other Polynomial functions Notes Quadra c func ons take the form f(x) = ax2 + bx + c The graph is a parabola which opens upward if a > 0, downward if a < 0. Cubic func ons take the form f(x) = ax3 + bx2 + cx + d. . Other power functions Notes Whole number powers: f(x) = xn . 1 nega ve powers are reciprocals: x−3 = 3 . √ x frac onal powers are roots: x1/3 = 3 x.. . General Rational functions Notes Deﬁni on A ra onal func on is a quo ent of polynomials. Example x3 (x + 3) The func on f(x) = is ra onal. (x + 2)(x − 1). . . 4.
5. 5. . V63.0121.001, Calculus I . Sec on 2.2 : Essen al Func ons . January 26, 2011 Professor Ma hew Leingang Outline Notes Algebraic Func ons Linear func ons Other Polynomial func ons Other power func ons General Ra onal func ons Transcendental Func ons Trigonometric Func ons Exponen al and Logarithmic func ons Transforma ons of Func ons Composi ons of Func ons. . Trigonometric Functions Notes Sine and cosine Tangent and cotangent Secant and cosecant. . Trigonometric functions graphed Notes GeoGebra applets. . . 5.
6. 6. . V63.0121.001, Calculus I . Sec on 2.2 : Essen al Func ons . January 26, 2011 Professor Ma hew Leingang Exponential and Logarithmic Notes functions exponen al func ons (for example f(x) = 2x ) logarithmic func ons are their inverses (for example f(x) = log2 (x)). . Graphs of exp and log Notes GeoGebra applets. . Outline Notes Algebraic Func ons Linear func ons Other Polynomial func ons Other power func ons General Ra onal func ons Transcendental Func ons Trigonometric Func ons Exponen al and Logarithmic func ons Transforma ons of Func ons Composi ons of Func ons. . . 6.
7. 7. . V63.0121.001, Calculus I . Sec on 2.2 : Essen al Func ons . January 26, 2011 Professor Ma hew Leingang Transformations of Functions Notes Take the squaring func on and graph these transforma ons: y = (x + 1)2 y = (x − 1)2 y = x2 + 1 y = x2 − 1 Observe that if the ﬁddling occurs within the func on, a transforma on is applied on the x-axis. A er the func on, to the y-axis.. . Vertical and Horizontal Shifts Notes Suppose c > 0. To obtain the graph of y = f(x) + c, shi the graph of y = f(x) a distance c units . . . y = f(x) − c, shi the graph of y = f(x) a distance c units . . . y = f(x − c), shi the graph of y = f(x) a distance c units . . . y = f(x + c), shi the graph of y = f(x) a distance c units . . .. . Now try these Notes y = sin (2x) y = 2 sin (x) y = e−x y = −ex. . . 7.
8. 8. . V63.0121.001, Calculus I . Sec on 2.2 : Essen al Func ons . January 26, 2011 Professor Ma hew Leingang Scaling and ﬂipping Notes To obtain the graph of y = f(c · x), scale the graph of f horizontally by c y = c · f(x), scale the graph of f ver cally by c If |c| < 1, the scaling is a compression If c < 0, the scaling includes a ﬂip. . Outline Notes Algebraic Func ons Linear func ons Other Polynomial func ons Other power func ons General Ra onal func ons Transcendental Func ons Trigonometric Func ons Exponen al and Logarithmic func ons Transforma ons of Func ons Composi ons of Func ons. . Composition is a compounding of Notes functions in succession g◦f x f . g (g ◦ f)(x) f(x). . . 8.
9. 9. . V63.0121.001, Calculus I . Sec on 2.2 : Essen al Func ons . January 26, 2011 Professor Ma hew Leingang Composing Notes Example Let f(x) = x2 and g(x) = sin x. Compute f ◦ g and g ◦ f. Solu on f ◦ g(x) = sin2 x while g ◦ f(x) = sin(x2 ). Note they are not the same.. . Decomposing Notes Example √ Express x2 − 4 as a composi on of two func ons. What is its domain? Solu on. . Summary Notes There are many classes of algebraic func ons Algebraic rules can be used to sketch graphs. . . 9.