Gr12 Mathematics: Ch 5 Sum Differential Calculus, Including Polynomials

What branch of mathematics is built on the concept of limits?

Trigonometry

Algebra

Calculus (correct)

Geometry

What is the paradox that illustrates the concept of limits?

The Speed of Light Paradox

Zeno's Speed Paradox

Achilles and the Tortoise Paradox (correct)

Achilles and the Hare Paradox

What is the value of y when x approaches -6 in the function y = (x^2 + 4x - 12)/(x + 6)?

-4

-6

-10

-8 (correct)

What is the graphical representation of the function y = (x^2 + 4x - 12)/(x + 6)?

A straight line with a hole at x = -6 Signup and view all the answers

Why can't we cancel the (x + 6) terms in the function y = (x^2 + 4x - 12)/(x + 6)?

Because the function is not defined when x = -6 Signup and view all the answers

What is the purpose of differential calculus?

To solve optimization problems and find rates of change Signup and view all the answers

What is the definition of the derivative of a function f(x)?

The derivative of a function f(x) is the limit as h approaches 0 of f(x + h) - f(x) over h. Signup and view all the answers

What is the general rule for differentiation of x^n?

The derivative of x^n is nx^(n-1) Signup and view all the answers

What is the derivative of a constant k?

The derivative of a constant k is 0 Signup and view all the answers

What is the derivative of a sum of two functions f(x) and g(x)?

The derivative of a sum is the derivative of f(x) plus the derivative of g(x) Signup and view all the answers

What is the notation for the derivative of a function f(x)?

All of the above Signup and view all the answers

What is the definition of a point of inflection?

The point where the graph changes concavity Signup and view all the answers

What is the use of the rules for differentiation?

To simplify the process of finding the derivative of a function Signup and view all the answers

What is the definition of the gradient of the tangent to a curve?

The gradient of the tangent to a curve is the derivative of the function Signup and view all the answers

What is the main application of differential calculus in optimization problems?

To find the stationary points of functions Signup and view all the answers

What is the derivative of a constant multiplied by a function f(x)?

k*f'(x) Signup and view all the answers

What is the formula for the remainder in synthetic division?

R = p(d/c) Signup and view all the answers

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

To determine the change in gradient of the original function Signup and view all the answers

What is the step to find the x-intercepts of a cubic polynomial?

Solve f(x) = 0 Signup and view all the answers

What is the derivative of a difference of two functions f(x) and g(x)?

The derivative of a difference is the derivative of f(x) minus the derivative of g(x) Signup and view all the answers

What is the relationship between the gradients of the tangent and the normal to a curve at a given point?

mtangent × mnormal = -1 Signup and view all the answers

When should you use the rules for differentiation?

When the question does not specify how to determine the derivative Signup and view all the answers

What is the definition of concave up?

The graph opens upwards Signup and view all the answers

How do you find the y-intercept of a cubic function y = ax^3 + bx^2 + cx + d?

Set x = 0 and solve for y Signup and view all the answers

What is the formula for long division of polynomials?

a(x) = b(x) * Q(x) + R(x) Signup and view all the answers

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

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

What is the application of differential calculus in rates of change?

To find the rate of change of functions Signup and view all the answers

What is the method to find the stationary points of a cubic polynomial?

Solve f'(x) = 0 Signup and view all the answers

What is the effect of a > 0 on the graph of a cubic function y = ax^3 + bx^2 + cx + d?

The graph rises to the right and falls to the left Signup and view all the answers

What is the step to find the y-intercept of a cubic polynomial?

Evaluate f(0) = d Signup and view all the answers

What is the purpose of finding the first derivative of a function f(x)?

To determine the rate of change of the function Signup and view all the answers

How do you find the x-intercepts of a cubic function y = ax^3 + bx^2 + cx + d?

Set y = 0 and solve for x Signup and view all the answers

What is the formula for synthetic division?

q2 = a3, q1 = a2 + q2 * d/c, q0 = a1 + q1 * d/c, R = a0 + q0 * d/c Signup and view all the answers

What is a local maximum of a cubic function?

A point where the function changes from increasing to decreasing Signup and view all the answers

What is the gradient of the tangent line to a curve at a point?

The derivative of the function at that point Signup and view all the answers

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

To find the local maximum and local minimum Signup and view all the answers

What is the remainder when dividing a polynomial p(x) by cx - d?

p(d/c) Signup and view all the answers

If p(x) = (cx - d) * Q(x), what is the degree of Q(x)?

one less than p(x) Signup and view all the answers

What is the condition for cx - d to be a factor of p(x)?

p(d/c) = 0 Signup and view all the answers

What is the expression for p(x) if cx - d is a factor of p(x)?

p(x) = (cx - d) * Q(x) Signup and view all the answers

What is the first step in solving a cubic equation using the Factor Theorem?

Use the Factor Theorem to find a factor Signup and view all the answers

What is the purpose of the Factor Theorem in solving cubic equations?

To factorize the polynomial Signup and view all the answers

What is the expression for the quadratic formula?

x = (-b + sqrt(b^2 - 4ac)) / 2a Signup and view all the answers

What is the relationship between the roots of a polynomial and its factors?

A polynomial has a root if and only if it has a factor Signup and view all the answers

What is the result of dividing a polynomial p(x) by cx - d if the remainder is zero?

p(x) = (cx - d) * Q(x) Signup and view all the answers

What is the final step in solving a cubic equation using the Factor Theorem?

Combine solutions from the quadratic polynomial Signup and view all the answers

What is the concept that the function y = (x^2 + 4x - 12)/(x + 6) illustrates?

The concept of limits Signup and view all the answers

What is the purpose of differential calculus in optimization problems?

To find both the maximum and minimum values of a function Signup and view all the answers

What is the graphical representation of the function y = (x^2 + 4x - 12)/(x + 6)?

A straight line with a hole at x = -6 Signup and view all the answers

Why can't we cancel the (x + 6) terms in the function y = (x^2 + 4x - 12)/(x + 6)?

Because x + 6 is only defined when x is not equal to -6 Signup and view all the answers

What is the main application of differential calculus?

All of the above Signup and view all the answers

What is the significance of the concept of limits in calculus?

It is the foundation of calculus, allowing us to study rates of change and accumulation Signup and view all the answers

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

To determine the rate of change of the function at that point Signup and view all the answers

What does the second derivative of a function indicate?

The change in gradient of the original function Signup and view all the answers

What is the relationship between the gradients of the tangent and the normal to a curve at a given point?

Their product is -1 Signup and view all the answers

What is the effect of a < 0 on the graph of a cubic function y = ax^3 + bx^2 + cx + d?

The graph falls to the right and rises to the left Signup and view all the answers

How do you find the x-intercepts of a cubic function y = ax^3 + bx^2 + cx + d?

By solving f(x) = 0 Signup and view all the answers

What is a local minimum of a cubic function?

A point where the function changes from decreasing to increasing Signup and view all the answers

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

To classify the stationary points as local maximum or local minimum Signup and view all the answers

How do you find the equation of the tangent line to a curve at a point?

By using the point-slope form of the equation of a straight line Signup and view all the answers

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

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

What is the purpose of differential calculus in optimization problems?

To find the stationary points of a function Signup and view all the answers

What is the definition of the derivative of a function f(x)?

The derivative of a function f(x) is written as f'(x) and is defined by the limit as h approaches 0 of f(x + h) divided by h. Signup and view all the answers

What is the notation for the derivative of a function f(x)?

f'(x), y', Dy, and Df(x) Signup and view all the answers

What is the general rule for differentiation of x^n?

The derivative of x^n is nx^(n-1), where n is a real number. Signup and view all the answers

What is the derivative of a constant k?

The derivative of a constant k is 0. Signup and view all the answers

What is the derivative of a sum of two functions f(x) and g(x)?

The derivative of a sum of two functions f(x) and g(x) is the derivative of f(x) plus the derivative of g(x). Signup and view all the answers

What is the use of the rules for differentiation?

The rules for differentiation are used to find the derivative of a function quickly and easily. Signup and view all the answers

What is the definition of the gradient of the tangent to a curve?

The gradient of the tangent to a curve is the derivative of the function at a point. Signup and view all the answers

What is the equation of a tangent to a curve?

The equation of a tangent to a curve is y = mx + b, where m is the gradient and b is the y-intercept. Signup and view all the answers

When should you use the rules for differentiation?

Whenever possible, to make finding derivatives faster and easier. Signup and view all the answers

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

f''(x) and D^2y Signup and view all the answers

What is the definition of a concave up curve?

A curve that opens upwards Signup and view all the answers

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

To find the points of inflection of a function Signup and view all the answers

What is the formula for the remainder in synthetic division?

R = p(d/c) Signup and view all the answers

What is the definition of a point of inflection?

A point where the curve changes concavity Signup and view all the answers

What is the purpose of the Factor Theorem in solving cubic equations?

To find the factors of the polynomial Signup and view all the answers

What is the condition for cx - d to be a factor of p(x)?

p(d/c) = 0 Signup and view all the answers

What is the graphical representation of a cubic function?

An S-shaped curve Signup and view all the answers

What is the purpose of differential calculus in optimization problems?

To find the maximum or minimum of a function Signup and view all the answers

What is the formula for long division of polynomials?

a(x) = b(x) * Q(x) + R(x) Signup and view all the answers

What is the method to find the stationary points of a cubic polynomial?

Solve f'(x) = 0 Signup and view all the answers

What is the remainder when dividing a polynomial p(x) by cx - d?

p(d/c) Signup and view all the answers

If p(x) = (cx - d) * Q(x), what is the degree of Q(x)?

One degree less than p(x) Signup and view all the answers

What is the condition for cx - d to be a factor of p(x)?

p(d/c) = 0 Signup and view all the answers

What is the expression for p(x) if cx - d is a factor of p(x)?

p(x) = (cx - d) * Q(x) Signup and view all the answers

What is the first step in solving a cubic equation using the Factor Theorem?

Identify a factor by trial and error Signup and view all the answers

What is the purpose of the Factor Theorem in solving cubic equations?

To find a factor Signup and view all the answers

What is the expression for the quadratic formula?

x = (-b (b^2 - 4ac)) / 2a Signup and view all the answers

What is the relationship between the roots of a polynomial and its factors?

A factor corresponds to a root Signup and view all the answers

What is the result of dividing a polynomial p(x) by cx - d if the remainder is zero?

p(x) = (cx - d) * Q(x) Signup and view all the answers

What is the final step in solving a cubic equation using the Factor Theorem?

Combine solutions Signup and view all the answers

Which of the following statements is true about the function y = (x^2 + 4x - 12)/(x + 6)?

The function has a limit as x approaches -6 Signup and view all the answers

What is the underlying concept behind Zeno's paradox of Achilles and the tortoise?

The concept of limits Signup and view all the answers

Why does the function y = (x^2 + 4x - 12)/(x + 6) have a hole at x = -6?

Because the denominator is zero at x = -6 Signup and view all the answers

What is the significance of the graph of the function y = (x^2 + 4x - 12)/(x + 6) in understanding limits?

It illustrates the concept of limits Signup and view all the answers

What is the relationship between the function y = (x^2 + 4x - 12)/(x + 6) and differential calculus?

The function illustrates the concept of limits, which is fundamental to differential calculus Signup and view all the answers

What is the underlying idea behind the development of calculus, including differential calculus?

The concept of limits Signup and view all the answers

What is the limit of the function y = (x + 6)(x - 2)/(x + 6) as x approaches -6?

-8 Signup and view all the answers

What is the derivative of the function f(x) = x^2 using the definition of a derivative?

2x + h Signup and view all the answers

What is the notation for the derivative of a function f(x) with respect to x?

All of the above Signup and view all the answers

What is the rule for differentiating a sum of two functions f(x) and g(x)?

The derivative of the sum is the sum of the derivatives Signup and view all the answers

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

All of the above Signup and view all the answers

What is the definition of the gradient of the tangent to a curve at a point?

The gradient of the tangent to the curve at the point Signup and view all the answers

What is the notation for the derivative of a function f(x) with respect to x, if y = f(x)?

All of the above Signup and view all the answers

What is the rule for differentiating a constant multiplied by a function f(x)?

The derivative is the constant times the derivative of the function Signup and view all the answers

What is the purpose of using the rules for differentiation?

To find the derivative of a function using shortcuts Signup and view all the answers

What is the derivative of the function f(x) = x^n, where n is a real number?

nx^(n-1) Signup and view all the answers

What is the relationship between the gradient of the tangent and the gradient of the normal to a curve at a given point?

Their product is -1 Signup and view all the answers

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

To determine the shape and orientation of the curve Signup and view all the answers

What is the effect of a < 0 on the graph of a cubic function y = ax^3 + bx^2 + cx + d?

The graph falls to the right and rises to the left Signup and view all the answers

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

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

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

To find the maximum or minimum values of the function Signup and view all the answers

What is the formula for finding the equation of the tangent line to a curve at a point?

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

What is the relationship between the gradient of the tangent and the y-coordinate of the point of tangency?

The gradient is independent of the y-coordinate Signup and view all the answers

What is the formula for finding the second derivative of a function f(x)?

f''(x) = d/dx[f'(x)] Signup and view all the answers

What is the purpose of finding the equation of the tangent line to a curve?

To describe rates of change Signup and view all the answers

What is the graphical representation of a local maximum of a cubic function?

A point where the function changes from increasing to decreasing Signup and view all the answers

If the second derivative of a function f(x) is positive at a certain point, what can be concluded about the function at that point?

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

What is the purpose of finding the second derivative of a function f(x) in the context of concavity?

To determine the concavity of the function. Signup and view all the answers

What is the remainder when dividing a polynomial p(x) by cx - d, according to the Remainder Theorem?

p(d/c) Signup and view all the answers

What is the condition for cx - d to be a factor of p(x), according to the Factor Theorem?

p(d/c) = 0 Signup and view all the answers

What is the expression for the quotient Q(x) when dividing a polynomial p(x) by cx - d, according to the Synthetic Division method?

p(x) / (x - d/c) Signup and view all the answers

If a cubic polynomial f(x) has a stationary point at x = k, what can be concluded about the polynomial?

The polynomial has a turning point at x = k. Signup and view all the answers

What is the purpose of finding the stationary points of a cubic polynomial f(x)?

To sketch the graph of the polynomial. Signup and view all the answers

What is the relationship between the first and second derivatives of a function f(x) at a point of inflection?

The first derivative is non-zero, and the second derivative is zero. Signup and view all the answers

What is the purpose of differential calculus in optimization problems?

To find the stationary points of a function. Signup and view all the answers

What is the application of differential calculus in rates of change?

To find the instantaneous rate of change of a function. Signup and view all the answers

If a polynomial p(x) is divided by cx - d and the remainder is R, what is the expression for p(x)?

p(x) = (cx - d)Q(x) + R Signup and view all the answers

What is the condition for cx - d to be a factor of p(x)?

p(d/c) = 0 Signup and view all the answers

If a polynomial p(x) has a root d/c, what can be said about cx - d?

cx - d is a factor of p(x) Signup and view all the answers

What is the degree of the quotient Q(x) when dividing a polynomial p(x) by a linear divisor cx - d?

One degree less than p(x) Signup and view all the answers

What is the first step in solving a cubic equation using the Factor Theorem?

Find a factor by trial and error Signup and view all the answers

What is the result of dividing a polynomial p(x) by cx - d if the remainder is zero?

cx - d is a factor of p(x) Signup and view all the answers

What is the expression for the quadratic formula?

x = (-b ± √(b^2 - 4ac)) / 2a Signup and view all the answers

What is the relationship between the roots of a polynomial and its factors?

The roots correspond to the factors Signup and view all the answers

What is the degree of the remainder R when dividing a polynomial p(x) by a linear divisor cx - d?

Zero Signup and view all the answers

What is the purpose of the Factor Theorem in solving cubic equations?

To factorize the polynomial Signup and view all the answers

Gr12 Mathematics: Ch 5 Sum Differential Calculus, Including Polynomials

Questions and Answers

What branch of mathematics is built on the concept of limits?

What is the paradox that illustrates the concept of limits?

What is the value of y when x approaches -6 in the function y = (x^2 + 4x - 12)/(x + 6)?

What is the graphical representation of the function y = (x^2 + 4x - 12)/(x + 6)?

Why can't we cancel the (x + 6) terms in the function y = (x^2 + 4x - 12)/(x + 6)?

What is the purpose of differential calculus?

What is the definition of the derivative of a function f(x)?

What is the general rule for differentiation of x^n?

What is the derivative of a constant k?

What is the derivative of a sum of two functions f(x) and g(x)?

What is the notation for the derivative of a function f(x)?

What is the definition of a point of inflection?

What is the use of the rules for differentiation?

What is the definition of the gradient of the tangent to a curve?

What is the main application of differential calculus in optimization problems?

What is the derivative of a constant multiplied by a function f(x)?

What is the formula for the remainder in synthetic division?

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

What is the step to find the x-intercepts of a cubic polynomial?

What is the derivative of a difference of two functions f(x) and g(x)?

What is the relationship between the gradients of the tangent and the normal to a curve at a given point?

When should you use the rules for differentiation?

What is the definition of concave up?

How do you find the y-intercept of a cubic function y = ax^3 + bx^2 + cx + d?

What is the formula for long division of polynomials?

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

What is the application of differential calculus in rates of change?

What is the method to find the stationary points of a cubic polynomial?

What is the effect of a > 0 on the graph of a cubic function y = ax^3 + bx^2 + cx + d?

What is the step to find the y-intercept of a cubic polynomial?

What is the purpose of finding the first derivative of a function f(x)?

How do you find the x-intercepts of a cubic function y = ax^3 + bx^2 + cx + d?

What is the formula for synthetic division?

What is a local maximum of a cubic function?

What is the gradient of the tangent line to a curve at a point?

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

What is the remainder when dividing a polynomial p(x) by cx - d?

If p(x) = (cx - d) * Q(x), what is the degree of Q(x)?

What is the condition for cx - d to be a factor of p(x)?

What is the expression for p(x) if cx - d is a factor of p(x)?

What is the first step in solving a cubic equation using the Factor Theorem?

What is the purpose of the Factor Theorem in solving cubic equations?

What is the expression for the quadratic formula?

What is the relationship between the roots of a polynomial and its factors?

What is the result of dividing a polynomial p(x) by cx - d if the remainder is zero?

What is the final step in solving a cubic equation using the Factor Theorem?

What is the concept that the function y = (x^2 + 4x - 12)/(x + 6) illustrates?

What is the purpose of differential calculus in optimization problems?

What is the graphical representation of the function y = (x^2 + 4x - 12)/(x + 6)?

Why can't we cancel the (x + 6) terms in the function y = (x^2 + 4x - 12)/(x + 6)?

What is the main application of differential calculus?

What is the significance of the concept of limits in calculus?

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

What does the second derivative of a function indicate?

What is the relationship between the gradients of the tangent and the normal to a curve at a given point?

What is the effect of a < 0 on the graph of a cubic function y = ax^3 + bx^2 + cx + d?

How do you find the x-intercepts of a cubic function y = ax^3 + bx^2 + cx + d?

What is a local minimum of a cubic function?

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

How do you find the equation of the tangent line to a curve at a point?

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

What is the purpose of differential calculus in optimization problems?

What is the definition of the derivative of a function f(x)?

What is the notation for the derivative of a function f(x)?

What is the general rule for differentiation of x^n?

What is the derivative of a constant k?

What is the derivative of a sum of two functions f(x) and g(x)?

What is the use of the rules for differentiation?

What is the definition of the gradient of the tangent to a curve?

What is the equation of a tangent to a curve?

When should you use the rules for differentiation?

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

What is the definition of a concave up curve?

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

What is the formula for the remainder in synthetic division?

What is the definition of a point of inflection?

What is the purpose of the Factor Theorem in solving cubic equations?

What is the condition for cx - d to be a factor of p(x)?