Gr12 Mathematics: Ch 5.5 Second Derivative

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What does the second derivative of a function indicate?

The gradient of the original function

The rate of change of the original function

The change in gradient of the original function (correct)

The maximum or minimum value of the original function

What is the notation for the second derivative of a function y?

dy/dx

d^2x/dy^2

y'' (correct)

y'

What is the purpose of finding the derivative of a function?

To find the volume of a solid

To find the area under a curve

To find the equation of a tangent line (correct)

To find the y-intercept

What is the formula to find the equation of the tangent line to f(x) at x = a?

y - y1 = m(x - x1) (A) Signup and view all the answers

What is the rule to find the second derivative of a function f(x)?

Differentiate f'(x) once (D) Signup and view all the answers

What is the purpose of the second derivative?

To determine the concavity of the graph (C) Signup and view all the answers

What is the relationship between the first and second derivative?

The second derivative is the derivative of the first derivative (B) Signup and view all the answers

What is the name of the function that has the form f(x) = ax^3 + bx^2 + cx + d?

Cubic function (B) Signup and view all the answers

What is the purpose of finding the second derivative of a function?

To determine the rate of change of the gradient of the function (D) Signup and view all the answers

What does the sign of the second derivative of a function indicate?

Whether the gradient of the function is increasing, decreasing, or remaining constant (B) Signup and view all the answers

What is the process to find the equation of the tangent line to f(x) at x = a?

Find the derivative at x = a, then use the point-slope form of the equation of a straight line (C) Signup and view all the answers

What is the relationship between the first and second derivative of a function?

The second derivative is the derivative of the first derivative (B) Signup and view all the answers

What type of function has the form f(x) = ax^3 + bx^2 + cx + d?

Cubic function (B) Signup and view all the answers

What is the notation to represent the second derivative of a function f(x)?

f''(x) (D) Signup and view all the answers

What is the purpose of finding the stationary points of a function?

To identify the points where the gradient of the function is zero (D) Signup and view all the answers

What is the use of the derivative in describing rates of change?

To describe the rate at which the output of the function changes with respect to its input (D) Signup and view all the answers

If the second derivative of a function is zero at a point, what can be concluded about the original function at that point?

The function has a constant gradient. (B) Signup and view all the answers

What is the geometric significance of the second derivative of a function at a point?

It represents the concavity of the graph. (D) Signup and view all the answers

If the second derivative of a function is positive at a point, what can be concluded about the original function?

The function is concave up. (C) Signup and view all the answers

What is the relationship between the first and second derivatives of a function at a stationary point?

The first derivative is zero and the second derivative is non-zero. (D) Signup and view all the answers

What is the purpose of finding the second derivative of a cubic function?

To determine the concavity of the graph. (C) Signup and view all the answers

If the second derivative of a function is negative at a point, what can be concluded about the original function?

The function is concave down. (D) Signup and view all the answers

What is the alternate notation for the second derivative of a function y?

d^2y/dx^2 (A) Signup and view all the answers

What is the role of the second derivative in analyzing the graph of a function?

It helps determine the concavity and inflection points. (B) Signup and view all the answers