NOTES : LIMITS AND DERIVATIVES Name: Date: Period: Mrs. Nguyen s Initial: LESSON.1 THE TANGENT AND VELOCITY PROBLEMS Pre-Calculus Mathematics Limit Process Calculus The type of it that is used to find TANGENTS and VELOCITIES gives rise to the central idea in DIFFERENTIAL CALCULUS, the DERIVATIVE. Tangent : Latin word tangens means touching. Tangent line should have the same direction as the curve at the point of contact. (Definition of tangent in Euclidean Geometry is inadequate.) Eample: The Tangent Line Problem P (c, f(c)) Q c, f ( c ) Slope of PQ = = f ( c ) f ( c) c c f ( c ) f ( c) Slope of a Tangent Line at point c, f ( c ) : f ( c ) f ( c) 0 Slope of the tangent line is the it of the slopes of the secant lines. Symbolically: m QP PQ m Mr. Farrar AP Calculus AB Chapter Notes Page 1
Practice Problem 1: Find the equation of the tangent line using the it process to the graph of the function at a given point c, f ( c ). f ( ) 3 Limit process 0 f ( c ) f ( c) a. 1, b., c., f ( ) Rates of Change i.e.: population growth, rates, production rates, water flow rates, velocity, and acceleration. Position Function The function(s) that gives the position (relative to the origin) of an object as a function of time t. Eample: The position of a free-falling object is represented by 1 where: s() t gt v0t s0 g = the acceleration due to gravity 3 ft / s 9.8 m/ s v 0 = initial velocity s = initial height 0 Mr. Farrar AP Calculus AB Chapter Notes Page
Average Velocity Instantaneous Velocity change in dis tan ce s change in time t s( t t) s( t) v( t) s'( t) t 0 t Note: Velocity is speed with direction Note: Instantaneous velocity function is the derivative of the position function Practice Problems: Mr. Farrar AP Calculus AB Chapter Notes Page 3
. Find the average velocity given: s t t t ( ) 16 16 3 a. 1, 3. Given: s t t t ( ) 16 16 3 a. When does the object hit the ground? b. 1,1.5 b. What is the object s velocity at impact? c. 1,1.1 Mr. Farrar AP Calculus AB Chapter Notes Page 4
4. Page 87 # t (min) 36 38 40 4 44 Heartbeats 530 661 806 948 3080 Estimate the heart rate between: a) 36,4 b)38,4 c) 40,4 d) 4,44 e) What are your conclusions? 5. Find an equation of the tangent line to 4 f ( ) 3 at 1 Mr. Farrar AP Calculus AB Chapter Notes Page 5
LESSON. THE LIMIT OF A FUNCTION Informal Description of a Limit If f( ) becomes arbitrarily close to a single number L as approaches c from either side, the it of f( ), as approaches c, is L. f ( ) L c Note: The eistence or non-eistence of f( ) at = c has no bearing on the eistence of the it of f( ) as approaches c. Eample: Given f ( ) 3, find the its as approaches 5. The Eistence of a Limit Let f be a function and let c and L be real numbers. The it of f( ) as approaches c is L iff f ( ) L and f ( ) L c c Eample: Infinite Limits Types of Limits f( ) or f( ) c We say: c the it of f( ) as approaches c is infinity (or negative infinity) or f( ) becomes infinite as approaches c or f( ) increases w/o bound as approaches c A real number L or Does not eist (DNE) Technically the it does not eist because is not a real number. However, the answer is either or. It s a particular way of epressing the it. Eample: Mr. Farrar AP Calculus AB Chapter Notes Page 6
Methods of finding its Direct substitution When Direct Substitution fails: Hole (or jump): 0 (indeterminate form) 0 - factor - cleanup fractions - multiply by the conjugate - swap trig identities - simplify logs - last resort: plug in # with tables and graphs Eamples: a) 5 3 5 5 Vertical Asymptote: # 0 - check both sides: possible or or DNE b) 3 5 5 Practice Problems: Find the its. 1. 5 3 3. 3 4 56 3. 1 4 8 3 4. 616 4 5. 5 4 3 65 15 6. 1 1 9 9 9 Mr. Farrar AP Calculus AB Chapter Notes Page 7
7. 3 3 5 6 3 8. h0 ( h) ( h) 5 ( 5) h 9. f ( ), where f ( ) 4 6, 4, 10. 3 1, 1 f ( ), where f ( ) 1 1, 1 11. csc 1. tan 13. ln 4 14. ln sin(4 ) 4 0 Mr. Farrar AP Calculus AB Chapter Notes Page 8
LESSON.3 CALCULATING LIMITS USING THE LIMIT LAWS Limit Laws Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following its. 1. Scalar Multiple bf ( ) f ( ) L and g ( ) K c c c bl. Sum or Difference ( ) ( ) c f g L K 3. Product f ( ) g( ) c LK Eample: Eample: Eample: 4. Quotient 5. Power f ( ) L, provided K 0 c g( ) K ( ) n n f L c 6. Horizontal Line 7. Line: y 8. Polynomial: 9. y b b c c c n y n c n c n n n c 10. Root Law Eample: Eample: Eample: Eample: Eample: c where n is a positive integer (if n is even, we assume c 0.) n f ( ) n f ( ) c c Practice Problems: Find the its if they eist. where n is a positive integer (if n is even, we assume f( ) 0.) c Eample: Eample: 1.. 4 16 4 Mr. Farrar AP Calculus AB Chapter Notes Page 9
3. sin 4. 0 4 1 cos The Squeeze Theorem If h( ) f ( ) g( ) for all in an open interval containing c, ecept possibly at c itself, and if h( ) L g( ) then f( ) c c c eists and is equal to L. Eample: c a b a f ( ) b a Find: f( ) c Practice Problems: Find the its. 5. Suppose Find 4 f( ) 0 and 4 f( ) f( ) 5, 4 6. 0 7. 0 5 7 sin 8. 0 3 5 7 sin 9. Suppose f( ). 1 f 6 ( ) 3 6, find 10. 0 3 1 cos 9 Mr. Farrar AP Calculus AB Chapter Notes Page 10
LESSON.5 CONTINUITY Definition of Continuity Continuity at a Point: A function f is continuous at c if the following three conditions are met. 1. f() c is defined.. f( ) c eists 3. f ( ) f ( c) c Note: Function f is continuous at c means that there is no interruption in the graph of f at c (no holes, jumps, or gaps). Continuity on an A function is continuous on an open interval ab, if it is continuous at each point in open interval the interval. A function that is continuous on the entire real line, is everywhere continuous. Discussion Most functions where we don t divide by zero are continuous on their domains Check when dividing by zero Polynomials are continuous everywhere Rational functions are continuous on their domains Inverse function of any continuous one-to-one function is also continuous Continuity of composite function: f ( g( )) f g( ) a a Discontinuity: if a function f is defined on an interval (ecept possibly at c) and f is not continuous at c Removable Discontinuity and Non-Removable Discontinuity Eample 1: f( ) 56 4 Eample : Describe any discontinuities 5 3 7 3 f ( ) 4 3 1 3 1 1 Mr. Farrar AP Calculus AB Chapter Notes Page 11
Practice Problems: Describe any discontinuities 1. 1 f( ) 1. 1 1 f ( ) 3 1 1 1 1 3. For what value(s) of the constant c is the function,? f( ) continuous on c 1 3 f( ) c 1 3 4. For what value(s) of the constant c is the function,? f( ) continuous on c f( ) c Intermediate Value Theorem ab, and k is any number between f( a ) and f() b,, that f () c k. If f is continuous on the closed interval then there is at least one number c in ab such Note: In other words, if takes on all values between a and b, f( ) must take on all values between f( a ) and f() b. Practice Problems: Use the Intermediate Value Theorem to show that f( ) has at least one zero on ab,. 5. f 3 ( ) 1, 0,1 6.,, 0 f 5 3 ( ) 3 4 7. f ( ) 5cos 4, 0, Mr. Farrar AP Calculus AB Chapter Notes Page 1
LESSON.6 LIMITS AT INFINITY; HORIZONTAL ASYMPTOTES Horizontal Asymptote i. n < m y =0 ii. n = m a y b iii. n > m no horizontal asymptotes n m Note: Horizontal asymptotes are in the form of y... Slant Asymptote When n > m by 1 the slant asymptote is the quotient of the rational epression. (If n > m by or more, the function has neither horizontal nor slant asymptotes) Note: Slant asymptotes are in the form y = m + b Limits at Infinity f ( ) L or f ( ) L Everything for larger then M fits inside the given window around L. Definition of a Horizontal Asymptote Limits at Infinity The line y L is a horizontal asymptote of the graph f is f ( ) L or f ( ) L. If r is a positive rational number and c is any real c number, then 0. r r Furthermore, if is defined when 0 c 0. r, then Remember: Given f ( ) L and c g ( ) K, then c f ( ) g( ) L K c f ( ) g( ) c In other words: cons tant 0 r cons tant 0 r LK or Mr. Farrar AP Calculus AB Chapter Notes Page 13
Practice Problems: Evaluate the following its and find their HA. 1. 3 3 7. 5 Note: when you encounter an indeterminate form, the suggested method is to DIVIDE the NUMERATOR and the DENOMINATOR by the HIGHEST POWER of in the DENOMINATOR. When asked to find all horizontal asymptotes, check both its as and. 3. 3 1 4. 3 5 3 1 5. 5 6 6. 6 4 Practice Problems: Evaluate the following its. 7. arctan( ) 8. e Mr. Farrar AP Calculus AB Chapter Notes Page 14
Functions with two horizontal asymptotes and nonrational functions. a. 3 1 Note: for 0 Eample 1: Find the following its for 0 b. 3 1 Graph: y 3 1 Functions with Infinite Limits at Infinity a. 4 1 Note: Eample : Find the following its f( ) b. 4 1 Graph: y 4 1 Be careful!!! 0 ( 1) Instead: Mr. Farrar AP Calculus AB Chapter Notes Page 15
LESSON.7 DERIVATIVES AND RATES OF CHANGE Definition of Tangent Line with Slope m Vertical Tangent Line The Derivative of a Function Definition of the Derivative of a Function Rates of Change Position Function Average Velocity If f is defined on an open interval containing c, and if the it y f ( c ) f ( c) m c, f ( c ) 0 0 eists, then the line passing through with slope m is the tangent line to the graph of f at the point c, f ( c ). If f is continuous at c and f ( c ) f ( c) or 0 f ( c ) f ( c), the vertical line, 0 c, passing through c, f ( c ) is a Vertical Tangent Line to the Eample: Eample: graph of f. Fundamental operation of calculus: Differentiation is the process of finding the derivative of a function. A function is differentiable at if its derivative eists at and differentiable on an open interval ab, if it is differentiable at every point in the interval. The derivative of f at is given by Notation for Derivatives: f ( ) f ( ) dy d f '( ) provided f '( ),, y ', f ( ), and D y 0 d d the it eists. For all for which this it eists, f ' is a function of. Note: The Derivative function is the Slope function of the original function. i.e.: population growth, rates, production rates, water flow rates, velocity, and acceleration. The function(s) that gives the position Eample: The position of a free-falling (relative to the origin) of an object as a object is represented by function of time t. 1 s() t gt v0t s0 where: g = the acceleration due to gravity 3 ft / s 9.8 m/ s v 0 = initial velocity s = initial height 0 Displacement change in dis tan ce s Note: Velocity is speed with directions Time change in time t Instantaneous s( t t) s( t) Velocity v( t) s '( t) t 0 t Note: Instantaneous velocity function is the derivative of the position function Mr. Farrar AP Calculus AB Chapter Notes Page 16
Practice Problems 1. Find the average rate of change of f ( ) 3 9,,4. Find the instantaneous rate of change of f ( ) 1 @ 4 3. Find the equation of the tangent line to f ( ) 4 5 6 @ 1 4. Find the equation of the tangent line to f ( ) @ 7 5. Given f ( ) cos, find f ' 6 6. Find the equation of the tangent line to 5 f ( ) @ 3 Note: cosh1 sinh 0 and 1 h0 h h0 h Mr. Farrar AP Calculus AB Chapter Notes Page 17
LESSON.8 THE DERIVATIVE AS A FUNCTION Definition of the Derivative of a Function Where might a function not have a derivative (not differentiable)? Higher-Order Derivatives Free-Falling Object Problems The derivative of f at is given by f ( ) f ( ) f '( ) provided 0 the it eists. For all for which this it eists, f ' is a function of. - Tangent line is vertical (undefined slope) - Corners (like absolute values) b/c of more than one tangent line @ a corner - Not continuous (jump) f ( ) function d dy f '( ) f ( ) y ' d d First derivative d d y f ''( ) f ( ) y '' d d Second Derivative 3 3 d d y f '''( ) 3 f ( ) y ''' 3 d d Third Derivative 4 4 (4) d d y 4 f ( ) 4 f ( ) y Fourth Derivative 4 d d n n ( n ) ( ) d ( ) d y n f f n y n d d th n Derivative Position Function st () Velocity Function v( t) s'( t) Notation for Derivatives: dy d f '( ),, y ', f ( ), and D y d d Note: The Derivative function is the Slope function of the original function. Graphs: Acceleration Function a( t) v'( t) s''( t) Eample 1: Prove f ( ) 4 is not differentiable at. Mr. Farrar AP Calculus AB Chapter Notes Page 18
Practice Problems: Find f '( ) using the definition. 1. 3 f ( ). 5 f( ) 3 3. f( ) 4 4. f( ) 4 Mr. Farrar AP Calculus AB Chapter Notes Page 19
Practice Problems: Given the following graphs, sketch the derivative. 5. 6. 7. 8. 9. 10. Mr. Farrar AP Calculus AB Chapter Notes Page 0