4.3 Worksheet - Derivatives of Inverse Functions

Transcription

1 AP Calculus 3.8 Worksheet 4.3 Worksheet - Derivatives of Inverse Functions All work must be shown in this course for full credit. Unsupported answers ma receive NO credit.. What are the following derivatives these MUST be memorized forwards (now) and backwards (later). a) d sin - = b) d - tan = c) d sec - =. Find the derivative of the following functions: - a) = tsin ( t ) b) ( ) - f = - + cos ( 3 ) 3. A particle is moving along the -ais so that its position at an time t > is given b (t). Find the velocit at the indicated value of t. Do without a calculator then check it with our calculator. - t a) () t = sin 4 - when t = 4. b) () t tan ( t ) = when t =.

2 p 4. Find an equation for the line tangent to the graph of = tan at the point (, 4 ). 5. Find an equation for the line tangent to the graph of = tan at the point (, p 4 ). 6. What is the relationship between the slopes of the tangent lines in questions 4 and 5? How does this help ou remember the rule for finding the derivative of an inverse function? 7. [Calculator Allowed] Let ( ) a) Find f () and f ' (). 3 f = + -. b) How can ou find ( 4) f - -? Find it. c) What is ( ) ( 4) f - -? d) What is ( f - ) ( ) 8. [Calculator Allowed] Suppose f ( ) a) Find f (4) and f '4 ( ) = b) What is ( ) ( 6) f -? c) What is ( ) ( 3) f - -?

3 ( ) - 9. The rule for inverse functions is that ( ) f f =. a) Take the derivative of both sides of the above epression. - b) Solve our equation from part a for the derivative of f ( ).. The functions f and g are differentiable for all real numbers and g is strictl increasing. The table below gives values of the functions and their derivatives at selected values of. g ( ) g '( ) If g - - is the inverse function of g, write an equation for the line tangent to the graph of g ( ) = at =.

4 AP Calculus 6. Worksheet Da 7. Worksheet # - Antidifferentiation b Substitution All work must be shown in this course for full credit. Unsupported answers ma receive NO credit. Indefinite Integrals: Remember the first step to evaluating an integral is to determine whether the integral is a form that should be recognized or whether u-substitution is needed.. 5. (-4 - ) sec e 6. e csc ( ) 7. sec ( ) cos( ) sec tan

5 3. - tan 4. csc ( 3 + 5) ( + + 7) 3 6. e tan( 3 ) 7. ( ) + e sec 3 e 8. ( + ) e e 9. What is the difference between definite and indefinite integrals? Use our prior knowledge of definite integrals to evaluate the following: cos( ) p. -p Let f be a differentiable function defined for all real numbers, with the properties listed below. Find f ( ). (i) f ( ) a b (ii) f () 6 and f () 8 (iii) f( ) 8

6 AP Calculus 6. Worksheet Da 7. Worksheet # - Antidifferentiation b Substitution All work must be shown in this course for full credit. Unsupported answers ma receive NO credit. p p True or False: tan ( ) sec ( ) u du Eplain our choice. While no one is going to force ou to do a definite integral problem using substitution a specific wa, the previous problem is less likel to be missed if ou get in the habit of changing the limits at the same time that ou make our substitution! e 4. ln e ( + 3) p sin

7 8. Algebraic Techniques To Integration Substitution (aka u-sub) works well when there is one part of the problem that is a derivative of the rest of the problem. When this doesn t occur, ou ma have to massage the problem to fit into a form that can be integrated from a rule or b using substitution. The more of these ou do, the better ou will get at recognizing which method will work. For now, use the following hints to help ou get started: a) Long Division You should use this when ou see b) Complete the Square You should use this when ou see c) Separate the Numerator You should use this when ou see d) Epand You should use this when ou see

8 3. 3 ( ) e e e e 3 7. ( )

9 9. [No Calculator] Consider a differentiable function f having domain all positive real numbers, and for which 3 f ' 4- - for >. it is known that ( ) ( ) a) Find the -coordinate of the critical point of f. Determine whether or not the point is a relative maimum, a relative minimum, or neither for the function f. Justif our answer. b) Find all intervals on which the graph of f is concave down. Justif our answer. c) Given that f () =, determine the function f. The following two integrals involve a twist to the normal substitution method. After ou make our normal substitution for u, ou have not accounted for all of the integrand replace the remaining s b solving our substitution rule for in terms of u.. ( )

10 AP Calculus 6. Worksheet Da 3 7. Worksheet #3 - Antidifferentiation b Substitution All work must be shown in this course for full credit. Unsupported answers ma receive NO credit.. Integrate each of the following: a) sin ( 4) b) 9+ 4 c) 9sec 7 d) ( 8tant + cos ( 8t) ) dt e) f) h) csc( 6) g) tan ( )

11 . [No Calculator] A graph of each function is given. Shade each region bounded b the graphs of the equations, then find the area of that region. a) + 4 p, =, = 4, and = b) sec 6, =, =, = Got substitution down? This question will determine how well ou trul understand EVERY aspect of the concept. Make our selection carefull. 3. If the substitution sin is made in the integrand of, the resulting integral is - A sin B sin C cos p 4 sin D p 4 sin E p 6 sin

12 4. Integrate each indefinite integral using an method possible. All Mied UP! a) cos( 3 ) sin( 3) e b) c) ln 3 ( ) ( ) ( ) sin 3 + cos 3 d) e) f) ( ) ( csc 3 cot 3 ) -9 g) - -e h) - + e - i)

13 5. Integrate each definite integral using an method possible. All Mied UP! a) - e b) c) ( ) [Optional] Need some more practice? From our tetbook Basic Integration Rules (p337 #-6) Substitution (p338 #5 4, 53 66)

14 AP Calculus 6. Worksheet 7. Worksheet - Slope Fields and Differential Equations All work must be shown in this course for full credit. Unsupported answers ma receive NO credit.. A general solution to a differential equation will have a in the solution.. Find the general solution to the differential equations below: (need more practice? page 37 # and #4) a) 5 sec sin - e b) - 3 c) ln d) ( ) 3. Find the particular solution = f () using the given initial condition. How are these different than solutions in the last question? (need more practice? p37 #,, 4, 6, 7, ) and = 3 when =. b) and = when t =. dt t t a) 4 c) du dv t t+ e + t and v = 5 when t =. dt 6 t and u = when =. d) 4sec tan 6

15 From our tetbook page 3 An initial condition determines a particular solution b requiring that a solution curve pass through a given point. If the curve is continuous, this pins down the solution on the entire domain. If the curve is discontinuous, the initial condition onl pins down the continuous piece of the curve that passes through the given point. In this case, the domain of the solution must be specified. 4. Which (if an) of the eamples in question 3, require ou to specif a domain? What are those domains? 5. Construct a slope field for each differential equation. Draw tin segments through the twelve lattice points shown in the graph. a) b) 6. For each slope field above, sketch the solution curve that passes through the point (, ). 7. [No Calculator] Consider the differential equation, where. a) On the aes provided, sketch a slope field for the given differential equation at the eight points indicated. b) Find the particular solution = f () to the differential equation with the initial condition f ( ) = and state its domain. O

16 8. Use separation of variables to solve each differential equation. Indicate the domain over which the solution is valid. a) and = when =. b) - and =.5 when =. Even though a differential equation can be given in a problem, sometimes we are not solving that differential equation tr #9. 9. Consider the differential equation a) On the aes provided, sketch a slope field for the given differential equation at the si points indicated. b) Find d in terms of and. c) Let f( ) be the particular solution to the differential equation with the initial condition f () 3. Does f have a relative minimum, a relative maimum or neither at =? Justif our answer. d) Find the values of the constants m and b for which m b is a solution to the differential equation.

17 d. Solve the initial value problem - 6 given that () = and ( ) ' 4.. [MATCHING] Connect each of the si slope fields shown below to their differential equations. Eplain each choice. - + cos + ( 3 ) -

18 AP Calculus 6.4 Worksheet 7.4 Worksheet - Eponential Growth and Deca All work must be shown in this course for full credit. Unsupported answers ma receive NO credit.. Suppose the rate of change of is proportional to the amount of present. a) Write the differential equation that this statement represents. b) Solve the differential equation from part a do not skip ANY steps.. Find the particular solution = f () to each differential equation using the given initial value. a) ( + 5)( + ) and = when =. b) ( ) 8 - and = 6 when = 5 c) cos and = when =. d) - e and = when =.

19 3. [No Calculator] The rate of change in the population of a group of elk in the local national forest is proportional to the difference between the maimum number of elk the forest can support and the number of elk currentl present. At time t =, when the number of elk are first counted, there are 4 elk. If L (t) is the number of elk at time t ears after the are first counted, then dl ( 5 L) dt 4 - a) Are the elk increasing in number faster when there are 6 or when there are 36? Eplain and use correct notation. b) Find an equation for d L dt in terms of L. What does d L dt tell ou about the graph of L? dl 5 - if L () = 4. dt 4 c) Use separation of variables to find the particular solution to ( L) 4. [No Calculator] If - and if = when t =, what is the value of t for which? dt A) - ln B) - 4 C) ln D) E) ln 5. [Calculator] A pupp weighs. pounds at birth and 3.5 pounds two months later. If the weight of the pupp during its first 6 months is increasing at a rate proportional to its weight, then how much will the pupp weigh when it is 3 months old? A) 4. pounds B) 4.6 pounds C) 4.8 pounds D) 5.6 pounds E) 6.5 pounds

20 6. [Calculator] During the zombie invasion of a small town the number of infected people is proportional to the difference between the town s population and the number of zombies currentl roaming around. There are 8 zombies roaming around when the are first discovered (call this time t = hours). If Z (t) represents the number of zombies roaming the town at time t, then dz.5( Z ) dt - a) Find a tangent line to the graph of Z when t =. b) Find dz dt in terms of Z. c) Use our tangent line from part a to estimate the number of zombies roaming the town 4 hours after the are first discovered (t = 4). Is this an over approimation or an under approimation? Eplain. d) Use separation of variables to find the particular solution for Z (t) if Z () = 8.