Worksheet on Derivatives. Dave L. Renfro Drake University November 1, 1999

Transcription

1 Worksheet on Derivatives Dave L. Renfro Drake University November, 999 A. Fun With d d (n ) = n n : Find y In case you re interested, the rimary urose of these roblems (Section A) is to review roerties of radicals and eonents.. y =. y =. y = q 4. y = y = q5 6. y = q 4 5 HINTS FOR (A). Don t you dare use the quotient rule!. If take your answer to # and raise it to the ower, then multily it by 9 9 6, you ll get the answer to #. 4. Don t you dare get a common denominator and combine these two terms before you di erentiate! 5. The di erentiation will be much easier if you rst searate out the constant factor: 5 = The di erentiation will be much easier if you rewrite 4 5 as (constant). B. Fun With Algebraic Functions: Find y. y = ( + ) +. y =. y = q a + b

2 4. y = q a b 5. y = a + a + a a + 6. y = + n + n + HINTS FOR (B). If you think for a moment you can avoid using the quotient rule.. You can use the roduct rule if you want, but here s a neat trick that s useful when something like this comes u in a ma/min roblem (chater 4 stu ): Make use of the fact that stu = 4 stu. 4. Begin by writing this as (constant) Before you di erentiate, rationalize the denominator. After you di erentiate and simlify you should get y = a + a a a. 6. After simlifying, the answer you should get is y = (n ) + + n. C. Fun With Transcendental Functions: Find y. y = e ln. y = (ln ) ln +. y = log 4 4. y = log 4 5. y = e e e 6 e 6 6. y = tan (5) tan (5) 7. y = ln Find + y (4) 9. y = ( ). y =. y =. y =. y = HINTS FOR (C)

3 . Simlify before you take the derivative!. This one should clean u very nicely after you simlify y.. Use the base change formula: log ( 4) = ln( 4) ln = ln ln ( 4). 4. Use the base change formula, followed by the quotient rule. 5. Don t you dare use the roduct rule on this one! 6. You can do this one in your head. (Doesn t this remind you of a trig. identity?) 7. Simlify before you take the derivatives! h 9. Note that ( ) = e ln ( ) i = e [ ln ]. Alternatively, you can write y = ( ), take ln (i.e. base e logarithm) of both sides, and then nd y by imlicit di erentiation.. Use the hint in # 9. [The answer IS NOT.]. Solve for y, then nd y. D. Fun With Imlicit Derivatives and Geometric Stu. Find the measures of the two airs of vertical angles, accurate to the nearest : sec, between y = and y = at the oint (; ).. Find the acute angle for the rst quadrant intersection of y = and y =. Give your answer in deg min sec, accurate to the nearest second.. Find the equation of the tangent line to y = + at =. Put your answer in y = m+b form. Grah the curve and its tangent line. Label the oint where = on your grah. 4. At what oint does y = ln increase the same rate as? At what oint does y = ln increase the same rate as? 5. Find y for each of the following: (a) y = + y (b) y = y (c) log y = log y 6. Find y at the oint ; if y + y = y 4.

4 7. Answer the following questions for the grah of y = +. (a) Find all the intercet(s) and all the y intercet(s). (b) Solve for y in terms of and use this to nd y as a function of. (c) Di erentiate y = + imlicitly, thereby nding y as a function of and y. (d) Check that your answers for (b) and (c) agree by relacing every aearance of y in your answer to (c) with what you found y to be equal to in terms of in the rst half of art (b). You should get the same eression that you got in the second half of art (b). (e) Find the equations of the tangent lines to this curve at each of its coordinate ais intercets (i.e. at each of the oints you found in art (a) ). (f) Find the equation of the tangent line to this curve at the oint whose is. coordinate (g) Find the equation of the tangent line to this curve at the oint in the second quadrant whose y coordinate is..5 y y = + 8. Find the sloe of ( + y ) = 8y at the following oints, accurate to 4 decimal laces. I suggest you start by nding signi cant digit aroimations for the coordinates of these oints and verify that they satisfy the equation I gave. Using the grah I ve rovided you can check to see if your aroimations to the coordinates of these oints seem reasonable and you can check that the sloes you nd at these oints seem reasonable. (a) ; (b) ; A 6 A 9 A = q

5 (c) 4 ; 4 8 (d) 4 ; B B B = q 4 + y ( + y ) = 8y HINTS FOR (D) 4. The rate of increase of a function f () is given by f (). 5. For (b), take ln (i.e. base e logarithm) of both sides, bring the eonent out, then di erentiate both sides. (Don t forget the roduct rule!) For (c), use the base change formula to convert each logarithm into base e logarithms, then di erentiate both sides. (Don t forget the quotient rule!) 6. Ste : Take the derivative of both sides. Ste : Take the derivative of both sides of what you got in ste. (You don t have to solve for y before taking the derivative again.) Use things like d d (y ) = y + y, etc. Ste : Plug = and y = into the equation you got in ste and solve for y. (You ll get a number.) Ste 4: Plug =, y =, and the value you got in ste for y into the equation you got in ste and solve for y. E. Fun With the Grah of y = sin. Grah y = sin on your calculator using the default (i.e. standard) window. Now Fact = 4 change the zoom factors to and zoom out once. Notice that the grah y Fact =

6 levels out to y when jj is large. Give a mathematically valid justi cation for why = using the squeeze theorem (also called the sandwich theorem). lim! sin. Notice that the grah of y = has a vertical asymtote at =. Since sin sin is a form as! [i.e. as!, both the numerator, and the denominator, sin, aroach ], it is not immediately obvious from the algebraic form of sin what lim is, or even if this limit eists. Multily the numerator and denominator! sin of by. This will give you a form as! (show details!), which means that the sin limit doesn t eist, the limit equals +, or the limit equals.. Notice that is an even sin function. Therefore, whichever of the three ossibilities DNE, +, holds for! + will also hold for! (why?). This means that we only need to look at what haens to as! sin +. We know that sin! as! +. There are three ways that we can have sin! : (a) sin aroaches from below, (b) sin aroaches from above, or (c) sin aroaches in an oscillatory fashion (i.e. as! +, sin is sometimes larger than and sometimes smaller than ). In a moment we will decide h i which case occurs. But for now, decide what lim! + sin = lim! + sin would have to be in each of these cases. 4. The grah of y = has sloe of at each oint. What is the sloe of y = sin when is just to the right of? Is this sloe always <, always >, or both, for all just to the right side of? What does your answer tell you regarding how fast the grah of y = sin rises just to the right of = as comared to how fast the grah of y = rises just to the right of =? Make sure that your conclusions are in agreement with a grahical check using your calculator. [Grah both y = and y = sin together and zoom in a little bit at (; ). Look to see which grah lies on to of the other, or if neither is always on to of the other, just to the right of =.] You should now be able to decide with near mathematical certainty which of (a), (b), or (c) in # occurs, and hence what lim! + sin is. 5. You may have noticed back in # that the grah of y = Look at the grah using these windows on your calculator: sin has lots of wiggles in it. (A) min = ma = and y min = : 8 y ma = : (B) min = ma = and y min = : 95 y ma = : 5 You may want to try some other windows, also. These just haen to be two that looked nice when I tried. Do these wiggles continue as!? The local maima and minima occur when y =. Find y and solve y =. You should be able to arrive at the equation sin cos =, which is equivalent to tan =. This equation has in nitely many Look at age 8 (bottom half) and age 9 (to half) of your tet if you need to refresh yourself on even functions. You ll want to know how to verify algebraically that a function is even and what symmetry roerty the grah of an even function has. Note that both sides of sin cos = are de ned for all real numbers, but tan isn t de ned when

7 solutions, as a grah of y = and y = tan together will show. You can nd these solutions by grahing y = tan and then zooming in at where y = tan crosses the ais. You will nd it much easier to inoint the locations of the intercets of y = tan than directly trying to inoint the locations of the maima and minima of the grah of y =. Using this as a hint, nd 4 decimal lace aroimations for the rst three sin relative maima in the grah of y = sin lying to the right of =. HINTS FOR (E). Your book aears to only mention the squeeze theorem on age 78. However, I went over the squeeze theorem back when I showed (in class) that lim sin! =. For the roblem at hand notice that sin always stays between and. Therefore, stays between and (why?). sin +. At some oint the fact that lim sin! = will be needed.. What does 6= and ositive something! + aroach? What about 6= and ositive something! 5. The rst three relative maima are near = 8, = 4, and =. or 6= and ositive something! ; +? F. Fun With Continuity and Di erentiability Conditions Answer questions (A) through (E) for the FIVE (5) grahs on the handout titled Continuity and Di erentiability by Insection. G. Fun With Related Rates Do roblems # 7,,, 6, 7 on age 4 of your tet. cos =. (Why?) However, none of the oints at which cos = can be a solution to sin cos = (Why?), and so rewriting as tan = doesn t cause us to overlook any solutions. [In contrast, note that by rewriting = 4 as = 4 we lose the = solution.]

8 H. Fun With Linear Aroimations Use an aroriate linear aroimation to aroimate the following:. 5:. 8: 8

9 8 >< ( + ) if < f () = ( >: + ) if < < ( + ) if >