Differentiation rules > Using the derivative in modeling
123456Using the derivative in modeling

Exercises

Exercise 1

Photos are printed on rectangular sheets of paper of 1 m2 to make posters. Around a picture the edge remains white: at the bottom edge a strip of 2 dm wide, at the other edges strips of 1 dm wide.
What dimensions of the poster maximize the printed area?

a

Make a sketch of this situation with the data added.

b

Try to solve the problem by yourself first. Only look at c and d if you don't succeed.

c

Assume that the width of a poster is denoted by x dm. Derive a formula for the area A of the printed part as a function of x .

d

Using differentiation calculate the value of x for which A ( x ) is maximal.

e

Now answer the question asked in the beginning.

Exercise 2

Somebody wants to buy a ladder to clean his gutters. Next to his house at a distance of 1 m from the wall there is a fence with a height of 3 m.

How tall should the ladder at least be so that it can be placed against the wall of the house over the fence?
(Assume that both the wall and the fence are perpendicular to the ground.)

Exercise 3

How long are the edges of the isosceles triangle with the greatest area and a circumference of 20 cm?

Exercise 4

Look at the graph of function f with f ( x ) = 10 - 2 x on the domain [ 0 , 5 ] . The line x = k (with 0 < k < 5 ) intersects the x -axis in point A and the graph of f in point B .

a

Show that the area A of rectangle O A B C is equal to: A ( k ) = k 10 - 2 k .

b

Use differentiation to determine what k maximizes the area of rectangle O A B C .

Exercise 5

Given the family of functions f p where f p ( x ) = x 2 + p x .

a

Algebarically calculate the extrema of f p when p = 1 .

b

For what values of p does f p not have extrema?

c

For what values of p does the tangent to the graph of f at x = 2 have a slope of -1 ?

Exercise 6

In a Cartesian coordinate system O x y line x = p with p > 0 intersects the graph of f ( x ) = 4 - x 2 in point P .

a

Calculate the minimal possible length line segment O P can have.

Now assume that 0 < p < 2 . Line x = p intersects the x -axis in A and rectangle A P Q B has points P and Q on the graph of f and point B also lies on the x -axis.

b

Calculate the maximum value for rectangle A P Q B .

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