Math 53 Second LE Reviewer

UP Academic League of Chemical Engineering Students (UP ALCHEMES) Academic Affairs Committee - Reviews and Tutorials Series, AY 2014-2015 Math 53 First Exam (REVIEWER) I. o

Introduction to derivatives: derivative as a slope of tangent lines

f

Suppose

lim

f ( x 1 +∆ x )−f ( x 1)

∴

eq’n of tangent line at

y−f ( x1 ) =m(x−x 1) 

x → ∆x

is

5.

Derivatives of Trigonometric Functions.

.

f ( x 1 +∆ x )−f ( x 1) ∆x

f −1

∆x

x →∆x

derivative of

f

The dom dom

f

±∞

is

x=x 1

, then,

.

defined by

f ( x 1+ ∆ x )−f ( x1 )

f −1 = lim



[ ]

P

eq’n of the tangent line The function

is called the

with respect to

f −1

Quotient Rule. VII. ' ' f ( x) g ( x ) f ( x )−f (x) g ( x) Dx = g (x) g (x)2

If the limit DNE, ie.,

lim

o

c.

,

is true.

∆x

x → ∆x

I.

P

is continuous at pt.

D x [ f ( x )∙ g(x ) ] =f ' ( x )+ g ' ( x)

VI.

x

6.

x

consists of all

b.

Dx [ cos x ] =−sin x

c.

Dx [ tan x ] =sec 2 x

d.

D x [ cot x ]=−csc 2 x

e.

Dx [ sec x ] =sec x tan x

f.

D x [ csc x ] =−csc x cot x

y=f ( g ( x ) )

If

for which the limit exists. III.

1. 2.

Dx [ c ] =0

cϵR

Power Rule.

+¿ n ϵ Z¿

dy dx

XII. Solve for

3.

cϵR

, V.

4.

f ,g

f

XIV.

x

x1

D x [ f ( x )± g( x )] =f ( x ) g ' ( x )+ g ( x)f ' (x ) Product Rule.

IV.

)(

(

XVII. Differentiability of a function XVIII.

Theorem: If then

f'

XIX.If

f

))

dy dy dy 2 − cot y+ x −csc y =2 x dx dx dx

dy 2 y− y 2 cos x+ cot y = dx 2 y sin x + x csc 2 y −2 x

XVI.

D x [ cf (x ) ] =c f ' (x)

are differentiable at

(

2

a.

b.

Solution: XV.

y cos x+sin x 2 y

is differentiable at

.

y 2 sin x−x cot y=2 xy

XIII.

D x [ n x n−1 ]

IV.

, then,

X. Implicit differentiation XI. Example:

x

II. Normal Line   to the T.L. III. II. Techniques of differentiation

u=g ( x )

and

dy dy du = = dx du dx

IX.

in the

( dxd , D )

D x [ sin x ] =cos x

Chain Rule. VIII.

.

a.

f

is differentiable at

is continuous at is defined at

differentiable at

x

, then:

x1

x1

.

, but not

x1

,

1.

f'

2.

f

3.

f'

is continuous (essential) at

'

has a vertical T.L at has no T.L at

x=x 1

x1

x1

XX. XXI. XXII. V. Rectilinear motion and derivatives as (Instantaneous) rates of change '

v ( t )=s (t)

1. XXIII.

XXV.

iii.

i.

VI.

- particle is moving to the

v <0

- instantaneous speed

a ( t )=v ' ( t ) =s ' ' (t)

2. XXIV. i. ii.

VII.

 

a<0

a<0

- particle is speeding up

v

and

a

have the same

x

close to

f

is differentiable at

x0

x0

.

,

f ( x ) ≈ f ( x 0 ) +f ' ( x 0 ) (x−x 0)

away and starts to rise at a rate of 10

v >0

a>0



XXX. Related rates XXXI. Example: XXXII. A boy observes a hot air balloon on the ground through a telescope. The balloon stands 100m

Remarks: A particle is If

- particle is slowing down

XXVI. Linear approximation and differentials XXVII.

XXIX.

left iii.

a>0

From (ii), if

For

- particle is moving to the

v =0



XXVIII. Suppose

right ii.

v <0

sign, then the particle is speeding up. Otherwise, the particle is slowing down.

Remarks:

v >0

If

m s

. At

what rate is the angle of elevation of the - particle is speeding up - particle is slowing down

telescope changing when the angle is

XXXIII.

dh m =10 dt s

dθ π =? at θ= dt 6 tan θ=

h dθ 1 dh sec 2 θ = 100 dt 100 dt

dθ 1 dh = cos 2 θ dt 100 dt dθ θ 1 3 3 rad at θ= = ∙ ∙10= dt 6 100 4 40 s

(

)

π 6

?