Average rates of change May be used to estimate the derivative at a point
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1 Derivatives Big Ideas Rule of Four: Numerically, Graphically, Analytically, and Verbally Average rate of Change: Difference Quotient: y x f( a+ h) f( a) f( a) f( a h) f( a+ h) f( a h) h h h Average rates of change May be used to estimate the derivative at a point Caution: Symmetric Difference Quotient as used on graphing technology may lead to incorrect answers. Instantaneous Rate of Change: Change in the dependent variable with respect to the independent variable Limit of the average rate of change y lim x 0 x Derivative of a function f at x = a is the limit of the Difference Quotient (this is a number): f( a+ h) f( a) f( x) f( a) f ( a) = lim f ( a) = lim h 0 h x a x a Slope of the curve at a point Derived function (Derivative) is the function defined by f ( x) = lim Differentiability Implies Continuity h 0 f( x+ h) f( x) h Displacement, Velocity, Acceleration, Speed (Speed is the absolute value of velocity) When is speed increasing? Tangent line: May intersect the function elsewhere May cross the function at the point of tangency Write the equation of the tangent (point slope form) y f( a) = f ( a)( x a) Write the equation of the normal Tangent line approximation: y = f( a) + f ( a)( x a) Units: Identify the units and explain using the correct units Implicit differentiation Mike Koehler - Derivatives
2 Table of Derivatives Power Rule d n u = nu dx n ( ) du dx Trigonometric Functions d du d du ( sin u) = cosu ( cosu) = sin u dx dx dx dx d du d du ( tan u) = sec u ( cot u) = csc u dx dx dx dx d du d du ( secu) = secutan u ( cscu) = cscucot u dx dx dx dx Inverse Trigonometric Functions d du d du ( sin u) = ( cos u) = dx u dx dx u dx d du d du ( tan u) = ( cot u) = dx + u dx dx + u dx d du d du ( sec u) = ( csc u) = dx u u dx dx u u dx Exponential Functions d u u du d u u du ( e ) = e ( a ) = a ln a dx dx dx dx Logarithmic Functions d ( ln u) = du d ( log ) du a u = dx u dx dx u ln a dx Product Rule: ( ) d uv = u dv + v du dx dx dx du dv v u d u Quotient Rule: dx dx = dx v v lo dhi - hi dlo lo Chain Rule: [ ] f( gx ( )) = f ( gx ( )) g ( x) dy dy du dy dy dt = = dx du dx dx dx dt Mike Koehler - Derivatives
3 AP Multiple Choice Questions 008 AB Multiple Choice Problems 6 00 AB Multiple Choice. If ( ) y = x +, then dy = dx A) ( ) x B) ( x + ) C) ( x + ) D) x ( x + ) E) 6x ( x + ). x + dy If y =, then = x + dx x + x A) B) (x + ) (x + ) C) (x + ) D) (x + ) E) 9. x If = ( + + ) f( x) ln x e, then f (0) is A) B) C) D) E) nonexistent. If y = x sin( x), then dy = dx A) xcos( x ) B) cos( ) x sin( x) xcos( x) x sin( x) + xcos( x) x x C) x( sin( x) + cos( x) ) D) ( ) E) ( ). Let f be the function defined by f( x) = x x+. Which of the following is an equation of the line tangent to the graph of f at the point where x =? A) y = 7x B) y = 7x+ 7 C) y = 7x+ D) y = x E) y = x 6. What is the slope of the line tangent to the curve y x = 6 xy at the point (, )? A) 0 B) 9 C) 7 9 D) 6 7 E) Mike Koehler - Derivatives
4 7. Let f be the function defined by f( x) = x + x. If g ()? A) B) C) 7 gx ( ) = f ( x) and g() =, what is the value of D) E) 998 AB Multiple Choice 8. π If f( x) = tan ( x), then f = 6 A) B) C) D) E) AB Multiple Choice 7. d cos dx ( x ) = A) 6x sin ( x ) cos( x ) B) 6x cos( x ) C) sin ( x ) D) 6x sin ( x ) cos( x ) E) sin ( x ) cos( x ) 7. If x A) d y + y =, what is the value of dx 7 B) 7 7 C) at the point (,)? 7 7 D) E) AB Multiple Choice 0. If A) = + +, then when x =, is y xy x B) dy dx C) - D) - E) nonexistent Mike Koehler - Derivatives
5 AP Free Response Questions 006 AB6 a b The twice-differentiable function f is defined for all real numbers and satisfies the following conditions: f(0) =, f (0) =, f (0) =. a) ax The function g is given by gx ( ) = e + f( x) for all real numbers, where a is a constant. Find g (0) and g (0) in terms of a. b) The function h is given by h( x) = cos( kx) f ( x) for all real numbers, where k is a constant. Find h ( x) and write an equation for the line tangent to the graph of h at x = AB Let f and gand their inverses f and g f, g, and the derivatives f and g at x = and x = be given in the table below. x f( x ) gx ( ) f ( x) g ( x) π 6 7 Determine the value of each of the following: a) The derivative of f + g at x = b) The derivative of f g at x = c) f The derivative of at x = g d) h () where hx ( ) = f( gx ( )) e) The derivative of g at x = Mike Koehler - Derivatives
6 Assessing Understanding of Speed. If velocity is negative and acceleration is positive, then speed is.. If velocity is positive and speed is decreasing, then acceleration is.. If velocity is positive and decreasing, then speed is.. If speed is increasing and acceleration is negative, then velocity is.. If velocity is negative and increasing, then speed is. 6. If the particle is moving to the left and speed is decreasing, then acceleration is. From Curriculum Module: Calculus: Motion available at AP Central Mike Koehler - 6 Derivatives
7 Textbook Calculus, Finney, Demanna, Waits, Kennedy; Prentice Hal, 0 Section Questions QQ p QQ p QQ p QQ p 8 Handouts The following pages contain handouts with problems accumulated from various sources. Mike Koehler - 7 Derivatives
8 Mike Koehler - 8 Derivatives
9 AP Calculus Chapter Section Sketch the graph of the derivative of the function whose graph is shown Mike Koehler - 9 Derivatives
10 Mike Koehler - 0 Derivatives
11 AP Calculus Chapter Section ANSWERS Mike Koehler - Derivatives
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13 AP Calculus Chapter Section. What is an equation of the tangent line at x =, assuming that f() = and f () =?. Suppose that y = x+ is the equation of the tangent line to the graph of y = f( x) at a =. What is f ()? What is f ()?. Suppose that f( x) is a function such that a) What is f '()? f( h) f() h h + = +. b) What is the slope of the secant line through (, f() ) and ( 6, f (6))? In exercises to 9, each of the limits represents a derivative f '( a ). Find f( x) and a. ( h) + x. lim. lim h 0 h x x π sin + h 6 6. lim x 7. lim h 0 x h x 8. lim 9. lim h 0 h x 0 + h x x ( + h ) 9 0. lim. lim x h 0 h x x. The cost of extracting T tons of ore from a copper mine is C = f( T) dollars. What does it mean to say f '(000) = 00? Mike Koehler - Derivatives
14 . The graph of the function f( x) is show at the right. Graph the function f '( x) on the same set of axes. 9. The graph y = f( x) is shown on the right. 8. For each of the following pairs of numbers, use the graph to decide which is larger. Explain your answer.. f() or f ()?. f() f() or f() f()? 6. f () f () () () or f f? 7. f '() or f '()? 8. Average rate of change between x = and x = or between x = and x =. 9. Arrange the following quantities in ascending order: 0 f '() f '() f() f() 0. Sketch the derivative of the graph of the function shown on the right. Sketch on the same axes. Mike Koehler - Derivatives
15 Answers y = ( x ) f '(000) = 00 says that when 000 tons of have already been extracted from the mine, the cost per ton is increasing at a rate of $00 per ton. f() = 7 f '() = f( + h) f() h + h f '() = lim = lim = h 0 h h 0 h f(6) f() f( + ) f() Slope of secant = = 6 h =, + = 7 f( x) = x a = f () f( x) = x a = f() f() 6 f( x) = sin( x) a = π 6 6 f() f() 7 f( x) = x a = 7 f '() 8 x f( x) = a = 8 Between and. 9 x f( x) = a = f '() f() f() f '() 0 f( x) = x a = 0 f( x) = a = x Mike Koehler - Derivatives
16 Mike Koehler - 6 Derivatives
17 AP Calculus Chapter Sections Work problems on a separate sheet of paper. Show all work. Sketch a picture where appropriate.. Given that the tangent line to y f( x) = at the point (, ) passes through the point ( ) 0,, find f ( ).. Given that f () = - and f '() =, find an equation for the line tangent to the graph of y = f( x) at the point where x =. x x. Let f( x) = ax + b x > Find the values of a and b so that f will be differentiable at x =.. The width of a rectangle is increasing at a rate of cm/sec, and its length is increasing at a rate of cm/sec. At what rate is the area of the rectangle increasing when its width is cm and its length is cm? (Hint: A= l w). Suppose f() =, f '() =, g() =, and g'() =. Find the derivative at of each of the following functions. a) px ( ) = f( x) + gx ( ) b) qx ( ) = f( xgx ) ( ) c) f( x) rx ( ) = d) gx ( ) sx ( ) = f( x) f( x) + gx ( ) f( x) 6. Let hx ( ) = f( xgx ) ( ), and jx ( ) =. Fill in the missing entries in the table below using the information gx ( ) about f and g given in the following table. x f( x ) f ( x) gx ( ) g ( x) h ( x) j ( x) / Mike Koehler - 7 Derivatives
18 7. Let f be a function defined by f ( x) = Ax + Bx + C with the following properties: ( ) i f(0) = ; (ii) f '() = 0; (iii) f ''(0) =. Find the values of A, B, and C. 8. Find c so that the line y = x+ is tangent to the curve y = x + c. 9. Find the x-coordinate of the point on the graph of cuts the curve at x = - and x =. y = x where the tangent line is parallel to the secant line that 0. Find the coordinate of all points on the graph of (,0). y = x at which the tangent line passes through the point. When an oil tank is drained for cleaning, there are Vt ( ) = 00, t+ 0t gallons of oil left in the tank t minutes after the drain valve is opened. (a) At what average rate does oil drain from the tank during the first 0 minutes? (b) At what rate does oil drain out of the tank 0 minutes after the drain valve is opened?. Find a function y = ax + bx + c whose graph has an x-intercept of, a y-intercept of, and a tangent line with a slope of at the y-intercept.. Find k if the curve y = x + k is tangent to the line y = 6x.. In 990, the population of the Unites States was about 0 million and was increasing at the rate of about million people per year. Per-capita income was about $,000 and was growing at about $000 per year. Both the U.S. population and U.S. per-capita income vary with time; their functions denoted by pt ( ) and it ( ) respectively. The product function wt () = ptit ()() describes the total annual U.S. income, another function of t. How fast was total annual income growing at time t = 990?. Suppose that f( x) is a function such that + =. What is f '()? f( h) f() h h Mike Koehler - 8 Derivatives
19 ANSWERS y+ = ( x ) a = 6 b = cm sec 7 7 a) b) c) d ) 6 6 x f( x ) f ( x) gx ( ) g ( x) h ( x) j ( x) / 7 A = B = C = 8 c = 7 9 x = 0 x = ± (.7, -.98) (.67(8),.98) (a) 00 gal or draining at a rate of 00 gal min min (b) V (0) = 00 gal min y = x x k = 9 w (990) = p(990) i (990) + i(990) p (990) = million, million 000 = billion + 0 billion = 9 billion f '() = Mike Koehler - 9 Derivatives
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21 AP Calculus Chapter Section.. Suppose f () = and the average rate of change of f between and is 6. Find f ().. Suppose that f () = and that f ( x) for all x in [ 0,0]. a) Could f () = 8? b) Could f ( 7) =? c) Could f (8) =?. Amy takes a trip from Chicago to Milwaukee. Due to road construction, she drives the first 0 miles at a constant speed of 0 mph. For the next 0 miles she maintains a constant speed of 60 mph and then stops at restaurant for 0 minutes for lunch. She drives the next miles at a constant speed of mph. a) Draw a graph that shows her distance along the road from Chicago as a function of time. b) Draw a graph that shows her velocity as a function of time. c) What is her average speed for the trip (including her stop at the restaurant)?. A car travels for 0 minutes with an average velocity of 0 mph, and then for 0 minutes with an average velocity of 0 mph. What is the average velocity of the car?. A car travels for 0 miles with an average velocity of 0 mph and then for 0 minutes at 60 mph. What is the average velocity of the car for the 60-mile trip? 6. A car is to travel miles. It goes the first mile at an average velocity of 0 mph. The driver wishes to average 60 mph for the entire two-mile trip. Is this possible? Explain. 7. Water is flowing into a spherical tank at a constant rate. Let Vt () be the volume of the water in the tank and Ht () be the height of the water level at time t. a) Give a physical interpretation of dv and dh. dt dt b) Is dv positive, negative or zero when the tank is one quarter full? Justify your answer. dt c) Is dh dt positive, negative or zero when the tank is one quarter full? Justify your answer. d) Which of dv dt and dh dt is constant? Explain your answer. 8. The position pt ()(in meters) of an object at time t (in seconds) along a line is given by pt () = t +. a) Find the change in position of the object between t = and t =. b) Find the average velocity of the object between t = and t =. c) Find the instantaneous velocity of the object at t =. d) Find the equation of the line tangent to the graph of pt () at t =. Mike Koehler - Derivatives
22 9. A cannon ball is shot upward with initial height feet and initial velocity 88 feet per second. This translates to about 00 mph, which is possible for an old-fashioned cannon. The height of the cannon ball is given by the equation ht ( ) = 6t + 88t+. Assuming free-fall conditions, answer the following questions. a) What is the average velocity from t = to t = 8 seconds? b) Find an expression for the velocity of the cannonball. c) What is the instantaneous velocity at t = 8 seconds? d) When does the cannonball reach its maximum height? How high does it rise? e) How long does the cannonball remain airborne? How fast is it going when it hits the ground? Answers 6 No, 7 is the maximum Yes, -6 is the minimum Yes 9. mph mph 8 mph 6 No. Last mile took two minutes. Must go entire trip in min. gal min ft min 7 Positive Positive dv dt m 8 m/sec 6 m/sec y = 6( x ) 80 ft./sec vt ( ) = t ft./sec 9 sec 0 ft sec and ft./sec Mike Koehler - Derivatives
23 AP Calculus Chapter Section. Work problems on a separate sheet of paper. Show all work. Sketch a picture where appropriate.. The height of a ball in feet t seconds after it is thrown is given by ht ( ) = 6t + 8t+ 7. a) From what height is the ball thrown? b) Find a function for the velocity of the ball. What was the ball's initial velocity? Was it thrown up or down? How can you tell? c) Was the ball's height increasing or decreasing at time t =? How fast was it moving? d) At what time did the ball reach its maximum height? How high was the ball at that time? e) How long was the ball in the air? How fast was it going when it hit the ground?. A particle moves along the x-axis so that its position in feet at any time t 0 is given by a) Find an expression for the velocity of the particle at any time t 0. b) Find the average velocity of the particle for the first two second. c) Find the instantaneous velocity of the particle at t = seconds. d) Find the values of t for which the particle is at rest. e) Find the position of the particle when it is at rest. f) Find an expression for the acceleration of the particle. When is the acceleration 0? g) Find the displacement of the particle from t = 0 to t = seconds. h) Find the total distance traveled by the particle from t = 0 to t = seconds. xt () = t t +.. A particle moves along the x-axis so that its position in meters at any time t 0 is given by xt ( ) = t 8t + t. a) Where is the particle at time t = 0? Where is the particle at time t =? b) Find an expression for the velocity at any time t 0. c) Find all values of t for which the particle is at rest. d) Find the total distance traveled by the particle from t = 0 to t = seconds.. A particle moves along the x-axis so that its position at any time t 0 is given by xt () = t t t+. For what values of t, 0 t is the particle's instantaneous velocity the same as its average velocity on the closed interval [0,]?. The position of a ball rolling down an inclined plane 8 meters long is given by the formula where s is the number of meters traveled after t seconds. st ( ). t.6t = +, a) How far has the ball traveled after seconds? b) How fast is the ball traveling after seconds? Indicate the units of measure. c) What is the average velocity of the ball on the interval from t = to t = seconds? d) Write an equation for v, the velocity of the ball at any time t, and use it to compute the velocity of the ball at the instant that it reaches the end of the inclined plane. Mike Koehler - Derivatives
24 6. A particle moves along the y-axis so that its position, measured in feet, at any time t 0 second is given by yt () = tsin() t + cos() t +. a) Find the position of the particle at time t = 0. b) Write an expression for the velocity of the particle in term of t. c) For what values of t, 0 t, is the particle moving upward? d) Write an expression for the acceleration of the particle in term of t. e) For t > 0, find the position of the particle the first time the velocity is zero. f) Find the total distance the particle travels over the interval 0 t. Answers a 7 feet.786 seconds b vt ( ) = t+ 8 8 ft sec up velocity postitive c Decreasing 6ft sec a meters d 8 =9 6 seconds 96.6 feet b. meters sec e.66 seconds ft sec c.6 meters sec d vt ( ) =. t+.6 v() =.6m sec a vt () = t t b ft sec 6a y (0) = ft c ft sec 6b vt () = tcos() t d At rest at 0 sec and sec 6c π π 0 < t < or < t seconds e Position is ft and ft 6d a = v = tsin( t) + cos( t) f at ( ) = t t= / =.77 sec 6e y π = π + =.7 feet g 6 feet 6f = feet h 6 feet a b c d meter -0 meter vt ( ) = t 6t+. sec. sec meters Mike Koehler - Derivatives
25 AP Calculus Chapter Section. Let hx ( ) = f( gx ( )). Use the information about f and g given in the table below to fill in the missing information about h and h'. x f( x ) f ( x) gx ( ) g ( x) hx ( ) h ( x). Assume that g is a function such that g ( x) exists for all x. Find the derivative of the function f. Answers will involve g and g. a. f( x) ( gx ( )) n n f( x) = g x c. f( x) = sin ( gx ( )) d. f( x) = g( sin( x) ) e. f( x) = tan ( gx ( )) e. f( x) = g( tan( x) ) = b. ( ). Use the figures below to evaluate the derivatives. a. c. d f ( gx ( )) dx = x d g ( f ( x )) dx = x b. d. d f ( gx ( )) dx = x 7 d g ( f ( x )) dx = x 7. Let hx ( ) = f( gx ( )) and jx ( ) = f( x) gx ( ). Fill in the missing entries in the table below. x f( x ) f ( x) gx ( ) g ( x) hx ( ) h ( x) jx ( ) j ( x) - 0 -/ 0 0 / - 0 -/ Mike Koehler - Derivatives
26 . Find the derivatives of y sin ( x ) and y sin ( x) 6. Assume that f(0) and f (0) 7. Find the derivative of y = tan( x) = =. = =. Find the derivatives of ( ) ( ) y = x + cos( x). 8. Find the derivative of ( ) 9. Find the derivative of y = tan ( x) + tan ( x ) f( x) and f 7 x at x= Some values of the derivative a function f are shown in the table below. No explicit formula for f is given. x 0 6 f ( x) a. Let gx ( ) = f( x+ ). For which values of x can you evaluate g ( x)? Evaluate g ( x) for these values. b. Let hx ( ) = f( x). For which values of x can you evaluate h ( x)? Evaluate h ( x) for these values. c. Let jx ( ) = f( x). For which values of x can you evaluate j ( x)? Evaluate j ( x) for these values. d. Let kx ( ) = f( x). For which values of x can you evaluate k ( x)? Evaluate k ( x) for these values.. Suppose that f is a function such that f ( x) 0 or greater than 0. Explain your answer. < for all x. If gx ( ) f( f( x) ) =, will g ( x) be less than, equal to,. If hx ( ) = f ( x) g( x), f ( x) = gx ( ), and g ( x) = - f( x), then h ( x) = A) 0 f( xgx ) ( ) B) f( xgx ) ( ) C) f( xgx ) ( ) D) 0 f( xgx ) ( ) E) f ( xg ) ( x) If gx ( ) = x, y= f gx ( ), and f ( x) = x, then dy = dx A) x x B) x x D) x x E) x x. ( ) x C) x Mike Koehler - 6 Derivatives
27 Answers x f( x ) f ( x) gx ( ) g ( x) hx ( ) h ( x) a. a n ngx ( ( )) g'( x) b n n g '( x )( nx ) c cos( gx ( )) g'( x ) d g'(sin( x))cos( x ) e sec ( gx ( )) g'( x ) f g'(tan( x))sec ( x ) c. - d. b. 0 x f( x ) f ( x) gx ( ) g ( x) hx ( ) h ( x) jx ( ) j ( x) - /0 0 -/. 0 0 / - 0 -/ xcos x sin( x)cos( x ) ( ) 6 6 dy sec x dx = 7 ( ) y = sin( x) x x + cos( x) 8 ( )( ) dy 9 ( x sec ( x ) sec ( x) tan ( x) ) 0 D A dx = + a) For 0,,, Values: 0,-,-6, and b) For 0,,,,,,6 Values:,-9, 6, 0,-,-8, c) For 0,, Values:, 0, d) For 0,,,,,,6 Values:,-,, 0,-,-6, Greater than zero. g ( x) = f ( f( x)) f ( x) ( )( ) = + Mike Koehler - 7 Derivatives
28 Mike Koehler - 8 Derivatives
AP Calculus AB: Semester Review Notes Information in the box are MASTERY CONCEPTS. Be prepared to apply these concepts on your midterm.
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