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Solved problems for the chapter:

Critical Points

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Applications of the derivative

1. Determination of tangent and normal to a curve:

1.1. Determining the tangent to a curve:

1.2. Determining the normal to a curve:

2. Using differentiation to calculate angles:

2.1. Angle between a curve and the abscissa axis:

2.2. Angle between a curve and the ordinate axis:

2.3. Angle between two curves

3. Determining the rise and fall of a function and its stationary points

3.1. Increasing and decreasing function:

3.2. Stationary points:

4. Extreme problems:

5. Forum

↓ Show theory [1] ↓

Detailed solution steps are available at:

https://en.openprof.com/wb/chapter:critical_points/4813/

Example 1.

Obtain the equation of a quadratic function that has for , the extreme value and the zero at .

Example 2.

If the functions is given as:

and

And function h is also given as:

Prove that
Write the domain of definition of the function h.
Calculate the limit
Prove that the function h is even.
Prove that the function h has no zeros.

Given the function f as:

Calculate the zero of the function f.
Calculate the poles of the function f.
Obtain the equation of the horizontal asymptote of the graph of the function f.
Calculate the derivative of the function f.
Draw the graph of the function in the given coordinate system.

Given the function f:

Draw the graph of the function
Write the intersections of the graph with the coordinate axes
Write the equations of the vertical and horizontal asymptotes.
Prove computationally that the function f has no stationary points.

Calculate the derivative of the function and obtain the stationary points.

Calculate the derivative of the function and obtain the stationary points:

Given the , find the zeros, intercept on the ordinate axis, stationary points and hence sketch the graph of the function.

Given the function , obtain the zeros, the intersection with the ordinate axis and the stationary points, and hence sketch the graph of the function.

Find two positive numbers whose sum is 14 and the sum of their squares is minimum.

10.

Given the function , obtain the zero(s), intersection with the ordinate axis, the point where the function is undefined and the stationary point.

11.

Sketch the graph of the function , showing the increasing and the decreasing intervals.

12.

Find the nature of the stationary points of the function .

13.

Calculate the nature of the stationary points of the function .

14.

Calculate the stationary points of the function .

15.

There is a dog kennel (1m x 1m) in a house. A 9m fence is bought to fence the kennel. What should be the length and width of the rectangular kennel so that the dog has the most space to move?

Space sketch:

16.

Make a box out of rectangular cardboard (80 cm x 60 cm) by cutting squares with the side x from the corners, folding the rest upwards and gluing them along the edges. Sketch of cardboard and boxes:

Calculate for what value of x will the volume of the box maximum?

17.

Find two positive numbers such that their sum is 50 and their product is the maximum.

18.

The patient receives medication. The rational function describes the concentration of the drug in the blood as a function of time t (time is measured in hours) after ingestion of the drug. When will the drug concentration in the blood be highest?

19.

Find the length of sides of a rectangle with perimeter P and has the largest area.