PROBLEMS FOR MATH 162 If a problem is starred, all subproblems are due. If only subproblems are starred, only those are due. SLOPES OF TANGENT LINES

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1 PROBLEMS FOR MATH 6 If a problem is sarred, all subproblems are due. If onl subproblems are sarred, onl hose are due. 00. Shor answer quesions. SLOPES OF TANGENT LINES (a) A ball is hrown ino he air. Is heigh afer seconds is meers. Wha does he slope of he secan line hrough wo poins (,( )), (,( )) on he graph of vs represen? Wha are is unis? (Answer: average change in posiion per ime inerval, over [, ], in m/s, also known as average veloci on his inerval.) Wha does he slope of he angen line a a poin (,()) on he graph represen? Wha are is unis? (Answer: insananeous change in posiion per ime inerval, over [, ], in m/s, also known as insananeous veloci a ime.) (b ) Waer drains from a ank. The ank holds a volume V of waer (in gallons) a ime (in minues). Wrie an expression for he slope of he secan line hrough poins (,V ( )), (,V ( )) on he graph of V vs. Wha does i represen? Wha are is unis? Wrie an expression for he slope of he angen line a he poin (,V ( )) on he graph, as a limi. Wha does i represen? Wha are is unis? (c ) The populaion of deer in an ecossem in ear is P (in housands of deer). Wrie an expression for he slope of he secan line hrough poins (,P( )), (,P( )) on he graph of P vs. Wha does i represen? Wha are is unis? Wrie an expression for he slope of he angen line a he poin (,P( )) on he graph, as a limi. Wha does i represen? Wha are is unis? LIMITS AND CONTINUITY 0. Find all verical asmpoes of he following funcions. Find he limiing behaviour of he funcion on eiher side of each asmpoe. Use his informaion o obain a skech of he graph of f(x) near he asmpoes. (a) f(x) = x x + 7 (b) f(x) = x x 7x + 0. For he funcion whos graph is shown in.5 #, sae all he poins x = a a which (a) he limi does no exis (b) he funcion is no coninuous DEFINITION OF DERIVATIVE 0. (a) Wrie a senence saing he mahemaical definiion of he derivaive of a funcion a x = a. (b) Wrie a senence saing an inerpreaion of he derivaive of a funcion a x = a. Be precise. (c) Under wha condiions is a funcion differeniable a a poin x = a? Wha can he graph of a funcion look like a a poin where i is no differeniable?

2 04. Fill in he blanks using eiher ma or mus. Explain our answer, possibl b giving example. (a) If f is differeniable a x = c hen i be coninuous a x = c. (b) If f is coninuous a x = c hen i be differeniable a x = c. RULES FOR DIFFERENTIATION 05*. Find he derivaive of f(x) = x + x + = (x + x + ) (a) using he quoien rule, (b) using he chain rule. Which wa is easier? 06*. Find he following derivaives. (a) Assuming = (x), find d dx ( ) (b) Assuming = (x), find d dx ( x) (c) Assuming u = u(), find d d (u ) (d) Assuming = (x) and a is a consan, find d dx (ax) One of he poins here is: ou need o know wha he independen variable is, wha he dependen variables are, and wha he consans are. 07. According o he heor of relaivi, he mass of an objec a veloci v is given b m(v) = m 0 v /c where c is he speed of ligh. (a) Find he domain of he funcion m(v). (b) Wha value does he mass approach as he paricles speed v approaches he speed of ligh (from below)? (c) Find he rae of change of an objec s mass wih respec o is veloci. (d) Wha are he unis of m(v)? Wha are he unis of m (v)? 08. The volume of a righ circular cone is V = πr h, where r is he radius of he base and h is he heigh. (a) Find he rae of change of he volume wih respec o he heigh if he radius is consan. (b) Find he rae of change of he volume wih respec o he radius if he heigh is consan.

3 LINEAR APPROXIMATIONS Noes: The linear approximaion of a funcion f(x) abou a base poin a is L(x) = f(a) + f (a)(x a) The graph of his linear funcion is he line angen o f a x = a. Read pp A useful applicaion is o approximae f b is linearizaion near a basepoin, f(x) L(x) if x a. For example sin x x if x 0 + x + x/ if x 0 The change in he funcion f beween x = a and x = a + x can be approximaed b he change in he linear approximaion L(x), f(a + x) f(a) L(a + x) L(a) ( ) This is useful since he righ hand side is paricularl simple o evaluae. Namel, if we denoe he change in f b, equaion (*) reduces o f (a) x ( ) Draw a picure showing and f (a) x. 09. Find he linear approximaion L(x) of he funcion f(x) a he given basepoin x = a. Skech a graph of boh f(x) and L(x). (a) f(x) = x 4 x, a =. (b) f(x) = x, a =. (c) f(x) = x, a = 0. (d) f(x) = an(x), a = Show ha for an real number k, ( + x) k + kx for small x. Esimae (.0) 0.7 and (.0) 0... The circumference c of a sphere was measured o be 84 cm wih a possible error of a mos 0.5 cm (in absolue value). Approximae he maximum error using his measuremen of c o compue (a) he surface area of he sphere (b) he volume of he sphere (Hin: in (a) he firs sep is o wrie surface area in erms of c. The wrie an approximaion of S in erms of c. Similarl for (b).). Approximae he amoun of pain needed o appl a coa of pain 0.05cm hick o a hemispherical dome wih diameer 50 m. Use linear aproximaions.. Approximae he volume of a hin clindrical shell wih heigh h, inner radius r, and hickness r. Use linear aproximaions. 4. The sopping disance for an auomobile (afer appling he brakes) is approximael F(s) =.s s f, where s is he speed in mph. Use he linear approximaion o esimae he change in sopping disance per addiional mph when s = 5 and when s = 55.

4 5. The reacion rae V of a common enzme reacion is given in erms of subsrae level S b V = V S K + S, S 0 where V > 0 and K > 0. Show ha V is an increasing funcion of S. Explain wh i follows ha V has no absolue maximum value. Wha is lim S V? Skech a rough graph of V as a funcion of S. 6. The engineer of a freigh rain needs o sop in 950 f o avoid sriking a barrier. The rain is raveling a a speed of 60 mi/hr. The engineer applies one se of brakes, which causes him o decelerae a a rae of 0. mi/min for 5 s. He realizes he is no going o make i, so he applies a second se of brakes, which causes he rain o decelerae a a rae of 0. mi/min. Will he srike he barrier? (Hin: careful, rounding can make he difference!) 7. Evaluae he following sums. 5 k (a) k (d) k= 000 (j) j= SUMMATION NOTATION (b) (e) 5 j sin(jπ/6) j=0 (c) 00 (m + m + 4) (f) m=0 00 k= N (m + m + 4), an N 8. Use summaion noaion o express he sum (a) (b) (c) /5 + /7 + 4/9 + 5/ 8 9. Which is larger, N j= j or N j= j? Explain wh. DEFINITE INTEGRAL 0. Use he definiion of he definie inegral o evaluae 0 x dx. (Tha is, firs wrie he inegral as he limi of a sum and hen evaluae he limi. You need o use he formulas 5-7 on page 0.). Le f() be he piecewise linear funcion whose graph is shown in he figures. Le F(x) = x 0 f()d Tha is, F(x) is he area under he graph of f beween = 0 and = x. The figures below show he area for 0 x <, x =, x >. Find a formula for F(x) in each of he hree cases. Is F(x) increasing as x increases? m=0 =f() =f() =f() x x< x= x x> 4

5 . Le f() be he funcion whose graph is shown in he figure a righ. Le F(x) = x f()d. On 0 wha inervals is F(x) increasing, and where is i decreasing? (All ou need here is o inerpre F(x) in erms of areas.) =f(). Show ha he funcion f(x) = x x + d, x > /, has an absolue minimum a x = + 5. Wrie is minimum value as a definie inegral (a) Skech he graph of f(x) = x. (b) Rewrie f(x) = x as a funcion defined piecewise. (c) Evaluae 0 x dx. INDEFINITE INTEGRAL 5. Find he following indefinie inegrals (a) sin(x) dx (b) d dx [sin(x )]dx (c) sec(πx) an(πx) dx 5