Introduction to Increasing and Decreasing Functions and Curve Sketching

The video tutorial covers curve sketching, focusing on understanding gradients and their role in determining increasing and decreasing functions. It explains differentiation and solving inequalities to find critical points. The tutorial also analyzes a complex function, y = x + 9/x, identifying stationary points and asymptotes. It concludes with determining ranges for increasing and decreasing functions, emphasizing graph sketching techniques.

What is the behavior of the curve y = x^3 - 3x^2 when x is between 0 and 2?

• The function is increasing.

• The function is decreasing.

• The function has no turning points.

• The function is constant.

What is the gradient of the function y = x^3 - 3x^2 after differentiation?

• 3x^2 - 6x

• x^3 - 3x^2

• x^2 - 3x

• 3x^3 - 6x^2

For the function y = x + 9/x, what are the x-values of the turning points?

• x = 0 and x = 2

• x = 2 and x = -2

• x = 3 and x = -3

• x = 1 and x = -1

What is the minimum point of the function y = x + 9/x?

• (2, 4)

• (0, 0)

• (-3, -6)

• (3, 6)

What is the range of x-values where the function y = x + 9/x is increasing?

• x < 3 and x > -3

• x < 0 and x > 2

• x > 3 and x < -3

• x > 0 and x < 2

What is the asymptote of the function y = x + 9/x?

• y = 0

• x = 3

• y = 3

• x = 0

Which of the following describes the behavior of the function y = x + 9/x between x = -3 and x = 3?

• The function is constant.

• The function is increasing.

• The function has no turning points.

• The function is decreasing.