Applications of Calculus Exercise

1. Find the rate of change of y with respect to x for the function y = x2 + 7x − 8 at the point x = 3.

2. Find the rate of change of volume with respect to time at the instant when t = 2 given that

V = 2t 2 − 5t + 8

where V is the volume of water leaking from a dam in m3 and t is the time in minutes.

3. Given that y = (x − 4)(x + 2)

a. Find the rate at which y is changing with respect to x when x = 3

b. For what value of x is the rate of change of y with respect to x equal to 16

4. Find the maximum area of a rectangle, given that its perimeter is 96 cm.

5. If x + y = 12, what is the minimum value of x2 + y2?

6. A rectangular paddock is to be fenced alongside a river. The side along the river does not need to be fenced. 600 metres of fencing is available for the other 3 sides. Let the width of the paddock be x metres.
What is the maximum area that can be fenced?

7. A sheet of cardboard is 90 cm by 60 cm. It is to be folded into an open box by cutting equal square of side x cm from each corner. a. How long is the box?

b. How wide is the box?

c. Find a formula for the volume of the box.

d. Find the maximum volume of the box. (You may need to use the quadratic formula!)

8. An object moves from a fixed point P in a straight line. Its velocity is given by the formula v = 3t- 4t + 10 m/s after t seconds.

a. Find an expression for the acceleration of the object.

b. What is the initial acceleration of the object?

c. Given that the object is at P when timing starts find an expression for s, the distance of the object from P after t seconds.

d. How far from the starting point is the object after 4 seconds?

9. A ball is thrown vertically into the air from ground level. Whilst the ball is in the air, its velocity is given by v = 36 − 6t m/s. Where t is the time in seconds.

a. What is the acceleration of the ball?

b. When will the ball be at its highest point?

c. When will it reach the ground?

10. The velocity of a particle is described by the relation v = 5t − 3, where t is the time in seconds.

a. If s in metres is the distance travelled from a fixed point and s = 0 when t = 3 find an expression for s.

b. Find the value of s when t = 5

c. What is the acceleration of the particle after 5 seconds?