AP Calculus Notes: Unit 1 Limits & Continuity. Syllabus Objective: 1.1 The student will calculate limits using the basic limit theorems.

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1 Syllabus Objective:. The student will calculate its using the basic it theorems. LIMITS how the outputs o a unction behave as the inputs approach some value Finding a Limit Notation: The it as approaches c o () ( ) I. Table sin E: 0 c Use the table to choose values o close to zero (rom the let and right). As approaches 0, it appears the unction is approaching. Note: Eistence at the point is not relevant when calculating a it. sin 0 Note: Special Limit must memorize! II. Graphically 4 E: Graph the unction and use the TRACE eature to see unction values as we approach rom the let and right. As approaches, it appears the unction is approaching Page o 6 Pearson Prentice Hall 007 Calculus: Graphical, Numerical, Algebraic..4 These notes are aligned to the tetbook reerenced above and to the College Board Calculus AB curriculum.

2 III. Analytically a. Direct Substitution 8 E: Substitute into the unction: 8 8 b. Algebraic Manipulation 9 E4: We cannot use direct substitution, because o division by zero. So we must simpliy the unction algebraically. 9 6 E5: 0 We cannot use direct substitution, because o division by zero. Simpliy the unction algebraically by rationalizing the numerator Noneistent Limit E6: We cannot use direct substitution, because o division by zero. Factoring the numerator, we have: This does not simpliy to avoid division by zero. So, does not eist. Page o 6 Pearson Prentice Hall 007 Calculus: Graphical, Numerical, Algebraic..4 These notes are aligned to the tetbook reerenced above and to the College Board Calculus AB curriculum.

3 One-sided Limits. Right-Hand Limit: the it o as approaches c rom the right. ( ) E7: Find, when Because we are approaching rom the right, we must substitute into the piece o the piecewise unctions that represents values o greater than. 5 Note: The it value diers rom the unction value at. c. Let-Hand Limit: the it o as approaches c rom the let. ( ) E8: We cannot use direct substitution, because we cannot approach rom the let. The domain o the unction is Two-sided Limit ( ) c,. So does not eist. ** has a it as approaches c i and only i the right and let hand its at c eist and are equal** Limits That Fail to Eist Right and Let Behavior Diers: c c Unbounded: or c c c Oscillating: For eample, sin. Show by graphing. 0 Page o 6 Pearson Prentice Hall 007 Calculus: Graphical, Numerical, Algebraic..4 These notes are aligned to the tetbook reerenced above and to the College Board Calculus AB curriculum.

4 E9: Use the graph below o to answer the ollowing questions. a. b. c. d. does not eist Page 4 o 6 Pearson Prentice Hall 007 Calculus: Graphical, Numerical, Algebraic..4 These notes are aligned to the tetbook reerenced above and to the College Board Calculus AB curriculum.

5 PROPERTIES OF LIMITS: I LMc,,, and kare real numbers and ( ) L and g( ) M, then c c. Sum and Dierence Rules: ( ( ) g( )) LM c The it o the sum or dierence o two unct ions is the sum or dierence o their its.. Product Rule: ( ( ) g( )) LM c The it o a product o two unctions is the product o their its.. Constant Multiple Rule: ( k ( )) kl c c The it o a constant times a unction is the constant times the it o the unction. () L 4. Quotient Rule:, M 0 c g () M The it o a quotient o two unctions is the quotient o their its, provided the it o the denominator is not zero. 5. Power Rule:I r and s are integers, s 0, then r r r s s s ( ( )) L provided that L is a real number. E0: Find tan. 0 Using direct substitution, we divide by zero. So try algebraic manipulation. We can rewrite tan as sin cos. sin tan cos sin sin cos 0 cos Using the product rule, sin sin 0 cos 0 0cos You Try: Calculate the it. t t t 4 QOD: Eplain how let and right-hand its relate to two-sided its. Page 5 o 6 Pearson Prentice Hall 007 Calculus: Graphical, Numerical, Algebraic..4 These notes are aligned to the tetbook reerenced above and to the College Board Calculus AB curriculum.

6 Sample AP Calculus AB Eam Question(s) (taken rom the released 00 MC AP Eam): For which o the ollowing does 4 eist? (Note: Each unction shown is.) I. II. III. (A) I only (B) II only (C) III only (D) I and II only (E) I and III only Page 6 o 6 Pearson Prentice Hall 007 Calculus: Graphical, Numerical, Algebraic..4 These notes are aligned to the tetbook reerenced above and to the College Board Calculus AB curriculum.

7 Syllabus Objective:. The student will calculate its using the basic it theorems. (Limits involving ininity.) Notation: E: Graph: increasingly ar to the right (or up) increasingly ar to the let (or down) ( ). Find and. 0 0 Deinition: The line y=b is a horizontal asymptote o the graph o a unction y = () i either ( ) b or ( ) b E: Find any horizontal asymptote(s) o the unction Find (or ). To calculate this it, begin by dividing each term by the highest 4 4 power o, which in this case is So the horizontal asymptote is y. Note: When inding g, you can use the ollowing guidelines. The g i the degree o is greater than the degree o g. Page 7 o 6 Pearson Prentice Hall 007 Calculus: Graphical, Numerical, Algebraic..4 These notes are aligned to the tetbook reerenced above and to the College Board Calculus AB curriculum.

8 The i the degree o is less than the degree o 0 g g. I the degree o is equal to the degree o g, then coeicients. g is equal to the ratio o the lead Ininite Limits Deinition: When approaches ininity as approaches a, the it does not eist, and is called unbounded. The line = a is a vertical asymptote o the graph o a unction y = () i either ( ) or ( ) a a E: Find the vertical asymptote(s) o 4. A raction will approach i the denominator o the raction approaches zero. To ind the vertical asymptote(s), set the denominator equal to zero and solve or. 40, Take a look at the graph: End Behavior Models: As becomes very large, a more complicated unction can be modeled by a simpler one. To see this, graph ( ) 5 6 and g( ) 4 4 Note: ( ) g ( ) For larger values o, the graphs appear alike. Page 8 o 6 Pearson Prentice Hall 007 Calculus: Graphical, Numerical, Algebraic..4 These notes are aligned to the tetbook reerenced above and to the College Board Calculus AB curriculum.

9 Finding End Behavior Models: Right End Behavior Model Find g such that ( ) g ( ) Let End Behavior Model Find g such that ( ) g ( ) E4: Find an end behavior model or End behavior model or the numerator: End behavior model or the denominator: 5 because because End behavior model or : 5 You Try: Find and when. QOD: Is an ininite it noneistent? Eplain your answer. Page 9 o 6 Pearson Prentice Hall 007 Calculus: Graphical, Numerical, Algebraic..4 These notes are aligned to the tetbook reerenced above and to the College Board Calculus AB curriculum.

10 Sample AP Calculus AB Eam Question(s) (taken rom the released 00 MC AP Eam):. For 0, the horizontal line y is an asymptote or the graph o the unction. Which o the ollowing statements must be true? (A) or all 0 (B) (C) (D) (E) 0 is undeined. 4 4 (A) 4 (B) (C) 4 (D) 0 (E) Page 0 o 6 Pearson Prentice Hall 007 Calculus: Graphical, Numerical, Algebraic..4 These notes are aligned to the tetbook reerenced above and to the College Board Calculus AB curriculum.

11 Syllabus Objective. The student will analyze the continuity o an elementary unction. Continuous Function (on an interval): a unction whose outputs vary continuously with the inputs and do not jump rom one value to another without taking on the values in between on the given interval (i.e. The graph can be traced without liting your pencil!) Continuous Function: a unction that is continuous at every point o its domain E: Graph the unction. Determine an interval in which the unction is not continuous. Is a continuous unction? In the interval,, the outputs do not take on all values in between and or any interval that includes = 0. So the unction is not continuous in the interval,. (Or any interval containing 0.) is a continuous unction, because it is continuous on every point in its domain. ( = 0 is not in its domain.) Deinition: Continuity at a Point. () is continuous at an INTERIOR POINT c o its domain i ( ) ( c ) c a. c is deined (eists) b. c eists c. ( ) ( c ) c. () is continuous at a LEFT ENDPOINT a or a RIGHT ENDPOINT b o its domain i ( ) ( a ) or ( ) ( b ) respectively. a b Page o 6 Pearson Prentice Hall 007 Calculus: Graphical, Numerical, Algebraic..4 These notes are aligned to the tetbook reerenced above and to the College Board Calculus AB curriculum.

12 E: Find the points o discontinuity o the unction shown in the graph. : ( ) does not eist because the let- and right-hand its are not equal. : Types o Discontinuity:. Removable (hole) 6 E: 4 at 4. Jump one-sided its eist, but are dierent (Greatest Integer Function), at every integer value o E4: Page o 6 Pearson Prentice Hall 007 Calculus: Graphical, Numerical, Algebraic..4 These notes are aligned to the tetbook reerenced above and to the College Board Calculus AB curriculum.

13 . Ininite (occurs when there is a vertical asymptote) E5: at y 4. Oscillating E6: sin at 0 Teacher Note: Have students graph on calculator and zoom in on = 0 to watch the oscillation. Removing a Discontinuity: a hole in a graph is a removable discontinuity 4 E7: Identiy the point o discontinuity in the unction. Then write an etended unction that is continuous at this point ( remove the discontinuity). Point o Discontinuity: 0 4 is the graph o 4 4 with a hole at. To ill the hole, we must ind 4 5. Etended unction: 4, 5, Page o 6 Pearson Prentice Hall 007 Calculus: Graphical, Numerical, Algebraic..4 These notes are aligned to the tetbook reerenced above and to the College Board Calculus AB curriculum.

14 Properties o Continuous Functions: I and g are continuous at = c, then the ollowing are continuous at = c. Sum: g Dierence: g Product: g Constant Multiple: k Quotient: g, provided gc 0, or any real number k Composite Functions: I is continuous at c and g is continuous at at c. c, then the composite g is continuous Intermediate Value Theorem: A unction value between a and b. y that is continuous on a closed interval, ab takes on every E8: The local weatherman reported that the low temperature today was 67 and the high was 89. From this inormation, can you guarantee that sometime during the day the temperature was 7? Eplain. YES. Temperature is continuous. It cannot jump rom one degree to another without hitting every value in between. So by the Intermediate Value Theorem, the temperature had to be 7 at some point during the day. You Try: Graph a unction or which eists, but is not continuous at. QOD: Can a continuous unction have a point o discontinuity? Eplain. Sample AP Calculus AB Eam Question(s): On which o the ollowing intervals is (A) 0, (B) 0, (C) 0, (D), (E), not continuous? Page 4 o 6 Pearson Prentice Hall 007 Calculus: Graphical, Numerical, Algebraic..4 These notes are aligned to the tetbook reerenced above and to the College Board Calculus AB curriculum.

15 Syllabus Objective: The student will calculate its using the basic it theorems. (Applications to instantaneous rate o change and slope) Average Rate o Change = amount o change time it takes (length o interval) Slope = y y m E: Find the average rate o change o ( ) over the interval [, ]. Finding the average rate o change is the same as inding the slope o the secant line. Average rate o change = 4 Pierre Fermat (69). Start with the slope o a secant through P and a point Q nearby.. Find the iting value o the secant slope as Q approaches P.. This is the slope o the curve at P and the slope o the tangent line to the curve at P. TANGENT TO A CURVE the line through P with the slope as calculated above The SLOPE OF THE CURVE y at the point, P a a is m h0 ( a h) ( a) h Note: As h approaches 0, the two points approach one point. The slope o the curve at point P is the same as the slope o the tangent line at point P. E: Find the slope o the parabola y at the point P(, 9). Write an equation or the tangent line at P. Slope: h ( h) () 9 6hh 9 6hh 6 h 6 h0 h h0 h h0 h h0 h h0 Equation o tangent line: y96 NORMAL LINE: the line perpendicular to a tangent line at a given point E: Write the equation o the normal line in the eample above. Perpendicular to the tangent at P: y9 6 Page 5 o 6 Pearson Prentice Hall 007 Calculus: Graphical, Numerical, Algebraic..4 These notes are aligned to the tetbook reerenced above and to the College Board Calculus AB curriculum.

16 INSTANTANEOUS SPEED: an object s speed at any given time E4: An object is dropped rom the top o a 50-t building. Its height above the ground ater t seconds is t. How ast is it alling second ater it is dropped? We must ind the instantaneous speed, or rate o change, at second h h4.9h h4.9h h h0 h h0 h h0 h h0 9.8 Note: Speed must be nonnegative. So the speed o the object at t = is 9.8 t/sec. You Try:. Find the average rate o change o the unction cos on the interval 0,.. Write the equations o the tangent and normal lines or at. y 4 QOD: How is the slope o a tangent line derived rom the slope o a secant line? Sample AP Calculus AB Eam Question(s) (taken rom the released 00 MC AP Eam): Let be the unction deined by 4 5. Which o the ollowing is an equation o the line tangent to the graph o at the point where? (A) y 7 (B) y 7 7 (C) y 7 (D) y 5 (E) y 5 5 Page 6 o 6 Pearson Prentice Hall 007 Calculus: Graphical, Numerical, Algebraic..4 These notes are aligned to the tetbook reerenced above and to the College Board Calculus AB curriculum.