ReviewProblemsforTest1
1.Let y = − x 2 + 4 .
a)Findtheslopeofthelinethroughthepoints ( 1 , 3 ) and ( 1+ h , f (1+ h) ) .
b)Whathappenstothisslopeashgetscloserandcloserto0?
2.Findeachofthefollowinglimits.Showallwork.
x 2 + x − 12
a) lim
x→ 3
x−3
2− x
b) lim
x→ 4
4−x sin ( 3θ )
c) lim
θ→ 0
θ
d) lim+
x→ 1
e) lim
x→ ∞
f) lim+
x→
π
2
x −1
x −1 3x 2 − 7
x2 + 4 sin x
cos x
g) lim
x→ ∞
h) lim−
x→ 1
5x + 9
x2 + 1 x+3
x2 − 1 3.Definepreciselywhatitmeansforafunctionftobecontinuous.Let
x2 − 4
f (x) =
for x ≠ 2 .Definefat x = 2 sothatfiscontinuousat x = 2 .
x−2
4.Whatcanyousayaboutthefunctionfwhosegraphisgivenbelow?
5.a)Definecarefully,intermsofalimit,thederivativeofafunctionfatthepoint
x = x0 .Itisdenotedby f '(x0 ) .
b)Whatarethetwoimportantinterpretationsof f '(x0 ) ?
c)Let y = f (x) = x 2 + 1 .Usethedefinitionofthederivativetofind f '(1) . 6.Findeachofthefollowingderivatives.Showallwork.
2
4
a) If f (x) = (3x + x − 1) , find f '(x) . (1− x )3 , find y' .
b) If y =
5
( x2 + 4)
2
c) If w = ln (1+ t ) , find w" . d 2u
−3x
, find
.
d) If u = x e
dx 2 dr
3
e) If r = sin ( 2θ ) , find dθ .
f ) If r = sin −1 ( 3θ ) , find
dr
.
dθ g) If w = x tan −1 (7x) , find w" .
7.Considerthecurvedefinedby ( x 2 + y 2 ) = 16xy .Findtheequationof
2
thelinetangenttothecurveatthepoint ( 2 ,2 ) .
8.Let f (x) = x 3 + 7 .Findthederivativeof f −1 at x = 8 . 9.Let y = ( sin x )x .Find y' .
10.Twocommercialplanesareflyingatanaltitudeof40,000feetalongstraight-line
coursesthatintersectatrightangles.PlaneAisapproachingtheintersectionpoint
ataspeedof442knots.PlaneBisapproachingtheintersectionat481knots.At
whatrateisthedistancebetweentheplaneschangingwhenplaneAis5nautical
milesfromtheintersectionpointandplaneBis12nauticalmilesfromthe
intersectionpoint?