calculus 3 tutorial 10 & 11

8/3/2019 CALCULUS 3 Tutorial 10 & 11

http://slidepdf.com/reader/full/calculus-3-tutorial-10-11 1/23

1

TUTORIAL 10

1. Find (a)

(b) (c)

Solutions:

(a)

—

(b)

(c)



2

2. Find .

(a) (b)

(c)

Solutions:

(a)

(b)

(c)



3

3. Find and

, assuming that y is a function of x.

(a)

(b)

(c)

Solutions:

(a)

(b)

(c)

*

+



4

4. Find the th derivative of the following functions, (a)

(b)

Solutions:

(a)

Derivatives Pattern

Therefore th derivatives



5

(b)

Derivatives Pattern

Therefore th derivatives



6

5. A curve has parametric equations , , for

Find the coordinates of the point at which the gradient of this curve is zero.

Solutions:

Given the gradient ,

When ,

Hence, the coordinates of the point is



7

6. The parametric of equations of a curve are , . Find and

as

functions of t .

Hence, show that

Solutions:

+ sin t

+

= 0 (shown)



8

7 . A curve has parametric equations . Find the equation of the tangent

to this curve at the point Solutions:

=

=

=

To find the value of , substitute the given point into equations (i) and (ii) : -

Since both equations get the same value of , the value of is omitted.

Substitute value of

into

:-

=

=

= 2

The equation of the tangent to the curve is: -

= 2

–



9

TUTORIAL 11

1. Find all critical numbers and the maximum and minimum values for f on the given

interval.

(a) (b) (c)

(d)

(e)

(f)

Solutions:

(a)

,

To find the minimum and maximum value,

When

When

When



10

(b)

When

To find minimum and maximum value:

When

When

When

(c)

When ,



11

To find the maximum and minimum value,

When

When

When

(d)

When

,


When

When

When



12

(e)

()

When ,


When

When

When

(f)

When ,



13


When –

When

–

When

–

When

–

2. Find the maximum and minimum values for on the given interval.

(a) (√ ),

(b) √ ,

Solutions:

(a) (√ )

(√ ) √

(√ )(√ )√

When ,(√ )(√ )√

(√ ) ( √ )

√ , √

√

√



14

When ;

(√ )

0

(√ )

(√ )

maximum value and minimum value .

(b) √

√ √

√

√

√

When ,

√

When :

√

√

√

maximum value and minimum value .



15

3. Find the numbers and if the function has the minimum value

on the interval .Solutions:

When , and at ,

----------------(i)

Given ,

---------(ii)

Substitute (i) into (ii) ;

and c .

4. Let Show that the function does not posses any local

extrema on the interval (,.

Solutions:

is defined for all x

72 and 42 are even numbers, thus for all x, x72 0 and x14 0 so that f ′( x) 4.

f does not have any critical number, hence it does not have any local extrema.



16

5. The sum of two nonnegative numbers is 20. Find these numbers if

(a) their product is as small as possible

(b) the sum of their squares is as small as possible

Solutions:

(a)

–

Product, =

= – ) , subject to and

=

= =

= =

–

–

–

Therefore the numbers are 0 and 20.

(b)

Sum,

=

=

Therefore, the numbers are both 10 and 10.



17

6. A rectangular play yard is to be constructed along the side of a house by erecting a

fence on three sides, using the house wall as the fourth side of the fence. Find the

dimensions that will maximize the play yard area if 20 m of fence is available for use.

Solutions :

x

y

When ,

[( )]

[( )]



18

7. An open box is to be made from a rectangular sheet of cardboard of dimension 16 cm

by 24 cm by cutting out squares of equal size from each of the four corners and

bending up the flaps. Find the dimensions of the box of largest volume that can be

made this way.

Solutions :

,

When

By using formula√

,

√



19

8. A rectangle has two vertices on the x-axis and the other two above the x-axis and on

the graph of the equation .Find the dimension for which the area of such

rectangle is a maximum.

Solutions :

Area, ------------------(i)

subject to

When ,

√

√ (omitted since out of interval)

When √ ,

√

Substitute √ and

into (i) :

√

√

√ √ √

√

The dimension for which the area of such rectangle is a maximum: √

)



20

9. A sector of a circle of radius r and angle , 0 ,is to have fixed perimeter P.

Find the dimension r and that maximize the area.

Solutions :

Perimeter

----------------(i)

When ,

Then, substitute into (i):

When

and



22

,

√ or √ (omitted since out of interval)

Thus, ( √ )

√ or √ (omitted since out of interval)

11. An orchard presently has 25 trees per acre. The average yield has been calculated to

be 495 apples per tree. It is predicted that for each additional tree planted per acre, theyield will be reduced by 15 apples per tree. Should additional trees be planted to

increase the yield? If so, how many trees should be planted to maximize the yield?

Solutions :

where x = no. of extra trees and its ranges

more plants should be planted to increase the yield



23

12. A hotel finds that it can rent rooms per day if it charges per room. For each

increase in rental rate, four fewer rooms will be rented per day. What room rate

maximizes revenues?

Solutions :

No. of room Rental rate per room

Normal

Increase $1,decrease 4 rooms –

To find the limit value of : -

–

So,

Let ,

–

– – – –

–

To find the maximum value ,

–

Test value of for and

[ – ]

[ – ]

Thus, the maximum increase in rental rate is so, the maximum rental rate is

per room.

calculus 3 tutorial 10 & 11

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