7 - 5fth week PDF

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5fth week ...

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Griffith University 1202SCG

1020ENG

TUTORIAL SHEET WEEK 7. CALCULUS 1 STUDENTS ONLY : Attempt questions 1(a), 3, 4, 6 and bring them with you to your tutorial. 1.

Sketch the curves of each of the following : (a)

y = x 4 - 12x 3 + 48x 2 - 64x ,

(

y = x 2 e! x ,

(c)

y = 2x 5 3 - 5x 4 3 ,

(d)

y = x3 -

3 2

2.

3.

x2 ,

(

2 x2 - 9

(e)

y=

(f)

y = x 4 - 2x 2 .

2

x -4

),

1 s = - gt 2 + v0 t + s0 where 2 g > 0 and constant, with s in metres and t in seconds. Here s0 is the initial height and v0 is the initial velocity of the body. Find the body’s maximum height.

The height of a body moving vertically is given by

A retailer has determined that the cost C of ordering and storing x units of a certain product is C = 2x +

300,000 , x

1 ≤ x ≤ 300.

The delivery truck can bring at most 300 units per order. Find the order size that will minimize the cost.

4.

A rectangular plot of farmland will be bounded on one side by a river and on the other three sides by a single-strand electric fence. With 800 m of wire at your disposal, what

5.

6.

The radius r of a circle is formula to estimate the increase in the area enclosed by the circle.

7.

(b)

Calculate the area enclosed by the circle when r = 10 m.

(c)

Using your answers to (a) and (b), estimate the area enclosed by the enlarged circle.

When a circular plate of metal is heated in an oven, its radius increases at the rate of 0.01 cm/min. At what rate is the plate’s area increasing when the radius is 50 cm ?

8.

9.

Water runs into a conical tank at a rate of 0.3 m3/min. The tank stands point down and has a height of 4 m and a base radius of 2 m. How fast is the water level rising when the water is 3 m deep ?

10.

French physiologist Jean Poiseuille discovered the formula to predict how much the radius of a partially clogged artery has to be expanded in order to restore normal flow. H

11.

A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius r of the outer ripple is increasing at a constant rate of 0.5 m/s. When the radius is 1 m, at what rate is the total area A of the disturbed water (the water enclosed by the outer ripple) changing ?

12.

Suppose that a drop of mist is a perfect sphere and that, through cond

13.

A piece of wire 80 cm long is cut into two pieces. Each piece is bent into the shape of a square. How should this be done to minimize th

14.

The profit P for a certain manufacturer selling x items is P(x) = 200x e- x/ 400 . Estimate the change in P as sales change from x = 145 to x = 150 items.

15.

When a shotputter projects a shot from height h above the ground at a speed v, its range R is given by

R=

v 2 cos ! g

" 2gh % 2 $# sin! + sin ! + v 2 '&

where ! is the angle of projection with the horizontal and g = 9.8m/s2. Find the angle ! that maximizes R. What is the angle when v = 13.7 m/s and h = 2.25 m ?...