Gradient and Tangent Concepts

The video tutorial explains how to measure the angle of inclination of a curve with respect to the x-axis. It begins by introducing the concept and then simplifies the problem by substituting the curve with a straight line. The tutorial uses trigonometry to find angles in right triangles and explains how to calculate gradients using calculus. The process involves using the derivative to find the gradient of the tangent at a specific point, which helps in determining the angle of inclination.

What is the main goal when measuring the angle a curve makes with the horizontal axis?

• To determine the curve's color

• To calculate the area under the curve

• To find the length of the curve

• To measure the angle of inclination

What is the relationship between the tangent of an angle and the sides of a right-angled triangle?

• Tangent is the sum of the sides

• Tangent is the difference of the sides

• Tangent is the ratio of opposite to adjacent sides

• Tangent is the product of the sides

If a line has a gradient of 1, what is the angle of inclination?

• 90 degrees

• 30 degrees

• 45 degrees

• 60 degrees

What happens to the angle of inclination as the gradient increases significantly?

• It approaches 0 degrees

• It approaches 90 degrees

• It remains constant

• It decreases to negative values

Why can't the tangent of a 90-degree angle be measured?

• Because it forms a straight line

• Because it is a right angle

• Because it is undefined in a triangle

• Because it is too small

What is the purpose of finding the gradient of a tangent at a point on a curve?

• To calculate the curve's length

• To measure the curve's width

• To find the angle of inclination

• To determine the curve's color

What mathematical tool is used to find the gradient of a tangent to a curve?

• Addition

• Derivation

• Multiplication

• Integration

How is the derivative of a function related to the gradient of a tangent?

• It is the average of the gradients

• It is the sum of the gradients

• It is unrelated to the gradient

• It is the gradient at a specific point

What is the derivative of the function f(x) = x^2 - x at x = 1?

• 1

• 0

• 2

• 3

What angle does the tangent make with the horizontal axis at x = 1 for the function f(x) = x^2 - x?

• 30 degrees

• 45 degrees

• 60 degrees

• 90 degrees