Maxine Lifshitz's Amsco's AP Calculus AB BC: Preparing for the Advanced PDF
By Maxine Lifshitz
Presents an entire overview of subject matters incorporated within the university Board AP Calculus AB and Calculus BC path descriptions and may support scholars boost the talents and techniques had to be successful on either the Calculus AB and Calculus BC assessments.
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Additional resources for Amsco's AP Calculus AB BC: Preparing for the Advanced Placement Exams
Note: Do not confuse fϪ1(x) with 1 . f(x) Use the horizontal line test to determine if a function is one-to-one from its graph. Horizontal Line Test If any horizontal line intersects the graph no more than once, then the function is one-to-one. If a function is not one-to-one, it may be possible to restrict its domain in order to make it one-to-one. This is the procedure for finding inverses of functions such as y ϭ x2 and y ϭ sin x. The graph of the inverse of a function is the reflection of the graph of the function (on its restricted domain) in the line y ϭ x.
If the graph is reflected across the x-axis, the new rule is y ϭ Ϫx2. When the graph of a parabola is shifted or reflected, the graph remains a parabola with properties similar to those of the original graph. If the graph of y ϭ ln x is shifted to the left 2 units, the new rule will be y ϭ ln(x ϩ 2). Practice in graphing both by hand and using a graphing calculator makes it easier to recognize that a group of functions can be understood as one function that has been shifted and/or reflected. EXAMPLE 1 Describe the translation and/or reflection that changes: (a) x2 to (x ϩ 3)2 (b) |x| to Ϫ|x Ϫ k| (k Ͼ 0) (c) 2x to 2Ϫx (d) cos x to Ϫcos x Solution (a) The graph is moved 3 units to the left.
Which of the following points is also on the graph of f ? (A) (Ϫ6, 5) (B) (6, Ϫ5) (C) (Ϫ6, Ϫ5) (D) (Ϫ5, Ϫ6) (E) (5, 6) f(x ϩ h) Ϫ f(x) 7. If f(x) ϭ "x Ϫ 2, then ϭ h (A) (B) (C) "x Ϫ 2 ϩ "h Ϫ 2 h "xh Ϫ 2 ϩ "x Ϫ 2 h "x Ϫ 2 ϩ h Ϫ "x Ϫ 2 h "x ϩ h Ϫ "2 h "x ϩ h Ϫ 2 Ϫ (x Ϫ 2) (E) h (D) 8. 3 Properties of Functions 27 (C) Ϫ1 (x ϩ h)(x ϩ h Ϫ 2) 1 h(x ϩ 2)(x ϩ h ϩ 2) 1 (E) 2h ϩ 2 9. Which of the following functions are odd? I y ϭ ln(x3) II y ϭ 0 x3 0 3 III y ϭ ex (D) (A) (B) (C) (D) (E) none II only I and II II and III I, II, and III 10.
Amsco's AP Calculus AB BC: Preparing for the Advanced Placement Exams by Maxine Lifshitz