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NATIONAL UNIVERSITY OF SINGAPORE

Department of Mathematics

MA 1505 Mathematics I

Tutorial 1

1. Find the first derivative of each of the following functions:

(a)ax + b

cx + d; (b) y = sinm x cos(nx), where m and n are constant positive integers;

(c) y =e4x + 5

e2x + 3; (d) y = [8x + ln (3 + 8x)]1/8, where x > 0.

Ans.

(a)

ad

bc

(cx + d)2 (b) sin

m1

x [m cos x cos(nx) n sin x sin(nx)]

(c)2e2x

e4x + 6e2x 5(e2x + 3)2

(d)4 + 8x

3 + 8x[8x + ln (3 + 8x)]7/8

2. Finddy

dxand

d2y

dx2for each of the following:

(a) x2/3 + y2/3 = a2/3, where a is a positive constant, 0 < x < a and y > 0;

(b) y = (ln x)lnx, where x > 1;

(c) x = a cos t, y = a sin t, where a is a positive constant.

Ans. (a) a

x

2/3

1 ; a2/3

3x4/3

a2/3 x2/3

(b) (ln x)lnx ln(ln x) + 1x

; (ln x)lnx

ln (ln x) + 1

x

2

+1 [ln (ln x) + 1] ln x

x2 ln x

(c) cot t ; 1a sin3 t

3. Sand falls from an open end of a pipe at the rate of 10 cubic metres per minute onto the top

of a conical pile. The height of the pile is always three-quarters of the base diameter. Find

the rate at which the base radius is changing when the pile of sand is 4 metres high.

Ans.15

16metres per minute

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MA1505 Tutorial 1

4. A police car chases a car of robbers on a straight southbound road. At a road intersection,

the robbers car turns left and heads down the straight eastbound road. When the police

car is still 0.3 km from the intersection on the southbound road, the radar in the police car

shows that the robbers car is 0.5 km away and the (instantaneous) rate of change of distancebetween the two cars is 0 km/hr. If the police car is travelling at 120 km/hr, find the speed

of the robbers car.

Ans. 90 km/hr

5. For each of the following functions f(x)

(a) f(x) =x + 1

x2 + 1, x [3, 3]; (b) f(x) = (x 1) 3

x2 , x (,)

find

(i) the values of x, if any, that give critical points, local extrema, and absolute extrema;

(ii) the (maximal) open intervals on which f(x) is (1) increasing, and (2) decreasing.

Ans.

(a) Critical points at x = 1

2; local max. at x = 3; local min. at x = 3;

absolute min. at x = 1

2; absolute max. at x = 1 +

2;

decreasing on (3, 1

2) and (1 +

2, 3); increasing on (1

2, 1 +

2)

(b) Critical points at x = 0 and x = 25

; local max. at x = 0; local min. at x = 25

;

no absolute extrema;

increasing on (, 0) and2

5,

; decreasing on

0, 2

5

6. A (circular) cylindrical container with no top cover is to be constructed to hold a fixed volume

V cm3 of liquid. The cost of the material used for the base is 8 cents/cm2, and the cost of

the material used for the curved surface is 3 cents/cm2. Find the radius r cm (in terms ofV)

of the least expensive container.

Ans.3

3V

8

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