Unit 5

More Applications of Derivatives

Slide 2

5.1 Optimization1) You wish to build a rectangular aquarium which holds 36 cubic feet of water.

You want the length to be twice the width. What dimensions would require the least amount of glass?

Slide 3

5.1 Optimization2) An offshore oil well is located at W, 5 miles from the closest point on shore

(A). The oil is to be piped from W to B, a point 8 miles down the shore from A, by piping it underwater from W to P and from P to B over land. If the cost of laying pipe under water is $100,000 /mile and the cost over land is $75,000/mile, where should P be located to minimize the cost of laying the pipe?

Slide 4

5.1 Optimization3) An open top box is to be made by cutting congruent squares of side length from the corners of a 20- by 25-inch sheet of tin and bending up the sides. How large should the squares

x be to make the box hold as much as possible?

What is the resulting volume?

Slide 5

5.1 Optimization2

4) A rectangle has its base on the -axis and its upper two vertices on the parabola 20 . What is the largest area the rectangle can have, and what are its dimensions?

xy x

xx

(x, 20 - x2)

Slide 6

5.1 Optimization7

5) How close does the curve come to the point , 0 ?2

y x

2

5

x, x

7

2, 0

Slide 7

5.2 Position, Velocity and Acceleration

Given position function s(t).

Velocity:

Speed:

Acceleration:

Slide 8

5.2 Position, Velocity and Acceleration

3 2

An object moves along a horizontal number line so that its positionat t seconds is 6 9 1.1) When is the object moving to the right?

2) Draw a motion diagram.

s tt tt

Slide 9

5.2 Position, Velocity and Acceleration

3 2

An object moves along a horizontal number line so that its positionat t seconds is 6 9 1.3) When is the acceleration negative?

4) What is the velocity when acceleration is zero?

5) What i

s tt tt

s the speed when acceleration is zero?

Slide 10

5.2 Position, Velocity and Acceleration

3 2

An object moves along a horizontal number line so that its positionat t seconds is 6 9 1.6) Draw a graph of on the interval 0, 6 ?

7) When is the object speeding up? slowing down?

s tt tts t

Slide 11

5.2 Position, Velocity and Acceleration

2

-2

5

Use the velocity graph to the right.8) When is the object moving forward?

9) When is the object moving backward?

10) When is the object speeding up?

11) When is the object slowing down?

12) When does the object attain its greatest speed?

Slide 12

5.2 Position, Velocity and Acceleration

2

-2

5

Use the velocity graph to the right.13) Graph the speed vs. time.

2

-2

5

Slide 13

5.3 Related Rates2 2If 2 1 differentiate with respect to time.x xy y

Slide 14

5.3 Related RatesSolving Related Rates Problems

1)

2)

3)

4)

5)

6)

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5.3 Related Rates1) A ladder 26 feet long leans against a vertical wall. The foot of the ladder slips

and it starts to slide down the wall. At the particular instant when the foot of the ladder is 10 feet from the base of the wall, find how fast the top of the ladder is moving down the wall. Assume the foot of the ladder is being drawn away from the wall at a constant rate of 4 feet per second.

Slide 16

5.3 Related Rates2) Suppose a liquid is being cleared of sediment by pouring it through a cone-

shaped filter. Assume the height of the cone is 16 inches and the radius of the cone is 4 inches. If the liquid is flowing out at a constant rate of 2 cubic inches per minute, how fast is the depth of the liquid decreasing when the liquid is 8 inches deep?

Slide 17

5.3 Related Rates3) An airplane flew over an airport at a rate of 300 miles/hour. Ten minutes later,

another plane flew over at 240 miles/hour. If the first plane was flying west and the second plane is flying south and both planes were flying at the same altitude, determine the rate at which they were separating 20 minutes after the second plane flew over the airport.

Slide 18

5.4 Linear ApproximationUsing a tangent line to f(x) at x = c to approximate values of f(x) for x – values close to c.

If f(x) is concave up (f”(x) > 0), then the approximation is an

.

If f(x) is concave down (f”(x) < 0), then the approximation is an

.

Slide 19

5.4 Linear Approximation1) Find the approximate value of at 9.2 from the tangent to the graph at 9.

y x xx

Slide 20

5.4 Linear Approximation 2) If 4 2sin , find the linear approximation of 0.1 .f x x f