Gr 11 Math Ch 5: Average Gradient

What is the average gradient of a curve between two points A and C?

The gradient of the curve at point C

The difference in y-coordinates of points A and C

The gradient of the tangent to the curve at point A

The gradient of the straight line passing through points A and C (correct)

What is the formula for the average gradient of a curve between points P(a, f(a)) and Q(a + h, f(a + h))?

(f(a + h) - f(a))/h (correct)

(f(a + h) - f(a))/a

(f(a + h) + f(a))/h

(f(a + h) × f(a))/h

What is the purpose of the concept of average gradient in calculus?

To describe the rate of change of a curve between two points (correct)

To calculate the gradient of a curve at a single point

To find the maximum value of a function

To find the equation of a curve

In the derivation of the average gradient formula, what happens to the value of h?

h approaches zero Signup and view all the answers

What is the gradient of the curve g(x) = x^2 at point P(a, g(a))?

2a Signup and view all the answers

What is the average gradient of the line AC with points A(x_A, y_A) and C(x_C, y_C)?

(y_A - y_C)/(x_A - x_C) Signup and view all the answers

What is the average gradient of the curve between points A and C, given points A(x_A, y_A) and C(x_C, y_C)?

(y_A - y_C) / (x_A - x_C) Signup and view all the answers

What does the average gradient of a curve represent?

The gradient of the line passing through two points Signup and view all the answers

Given points P(a, f(a)) and Q(a + h, f(a + h)) on a curve, what is the expression for the average gradient?

(f(a + h) - f(a)) / h Signup and view all the answers

In the derivation of the average gradient formula, what is the role of point Q(a + h, f(a + h))?

It is used to find the average gradient between P and Q Signup and view all the answers

What happens to the value of h in the average gradient formula?

It approaches zero Signup and view all the answers

What is the relationship between the average gradient and the gradient at a point?

The average gradient approaches the gradient at a point as h approaches zero Signup and view all the answers

What is the main difference between the gradient of a curve and the average gradient of a curve?

The gradient of a curve is the gradient of the tangent to the curve, whereas the average gradient is the gradient of the secant. Signup and view all the answers

Given a curve f(x) with points P(a, f(a)) and Q(a + h, f(a + h)), what is the expression for the average gradient in terms of a and h?

f(a + h) - f(a) / h Signup and view all the answers

If the average gradient of a curve f(x) between points P(a, f(a)) and Q(a + h, f(a + h)) is 2a + h, what is the curve f(x) equivalent to?

x^2 Signup and view all the answers

What is the significance of the point Q(a + h, f(a + h)) in the derivation of the average gradient formula?

It is the point used to calculate the average gradient over an interval. Signup and view all the answers

What happens to the value of the average gradient as the distance between points P and Q approaches zero?

The average gradient approaches the gradient of the curve at point P. Signup and view all the answers

Given a curve g(x) = x^2, what is the average gradient between points P(a, g(a)) and Q(a + h, g(a + h)) in terms of a and h?

2a + h Signup and view all the answers