Gr12 Mathematics: Ch 5 Sum Differential Calculus, Including Polynomials

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Questions and Answers

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)?

<p>A straight line with a hole at x = -6 (C)</p> 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)?

<p>Because the function is not defined when x = -6 (C)</p> Signup and view all the answers

What is the purpose of differential calculus?

<p>To solve optimization problems and find rates of change (D)</p> Signup and view all the answers

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

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

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

<p>The derivative of x^n is nx^(n-1) (D)</p> Signup and view all the answers

What is the derivative of a constant k?

<p>The derivative of a constant k is 0 (B)</p> Signup and view all the answers

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

<p>The derivative of a sum is the derivative of f(x) plus the derivative of g(x) (D)</p> Signup and view all the answers

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

<p>All of the above (D)</p> Signup and view all the answers

What is the definition of a point of inflection?

<p>The point where the graph changes concavity (B)</p> Signup and view all the answers

What is the use of the rules for differentiation?

<p>To simplify the process of finding the derivative of a function (A)</p> Signup and view all the answers

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

<p>The gradient of the tangent to a curve is the derivative of the function (B)</p> Signup and view all the answers

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

<p>To find the stationary points of functions (B)</p> Signup and view all the answers

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

<p>k*f'(x) (B)</p> Signup and view all the answers

What is the formula for the remainder in synthetic division?

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

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

<p>To determine the change in gradient of the original function (B)</p> Signup and view all the answers

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

<p>Solve f(x) = 0 (A)</p> Signup and view all the answers

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

<p>The derivative of a difference is the derivative of f(x) minus the derivative of g(x) (C)</p> 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?

<p>mtangent × mnormal = -1 (A)</p> Signup and view all the answers

When should you use the rules for differentiation?

<p>When the question does not specify how to determine the derivative (D)</p> Signup and view all the answers

What is the definition of concave up?

<p>The graph opens upwards (D)</p> Signup and view all the answers

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

<p>Set x = 0 and solve for y (C)</p> Signup and view all the answers

What is the formula for long division of polynomials?

<p>a(x) = b(x) * Q(x) + R(x) (B)</p> Signup and view all the answers

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

<p>f''(x) (B)</p> Signup and view all the answers

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

<p>To find the rate of change of functions (B)</p> Signup and view all the answers

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

<p>Solve f'(x) = 0 (D)</p> 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?

<p>The graph rises to the right and falls to the left (D)</p> Signup and view all the answers

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

<p>Evaluate f(0) = d (D)</p> Signup and view all the answers

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

<p>To determine the rate of change of the function (D)</p> Signup and view all the answers

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

<p>Set y = 0 and solve for x (C)</p> Signup and view all the answers

What is the formula for synthetic division?

<p>q2 = a3, q1 = a2 + q2 * d/c, q0 = a1 + q1 * d/c, R = a0 + q0 * d/c (C)</p> Signup and view all the answers

What is a local maximum of a cubic function?

<p>A point where the function changes from increasing to decreasing (A)</p> Signup and view all the answers

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

<p>The derivative of the function at that point (B)</p> Signup and view all the answers

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

<p>To find the local maximum and local minimum (D)</p> Signup and view all the answers

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

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

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

<p>one less than p(x) (C)</p> Signup and view all the answers

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

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

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

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

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

<p>Use the Factor Theorem to find a factor (A)</p> Signup and view all the answers

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

<p>To factorize the polynomial (D)</p> Signup and view all the answers

What is the expression for the quadratic formula?

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

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

<p>A polynomial has a root if and only if it has a factor (D)</p> 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>p(x) = (cx - d) * Q(x) (C)</p> Signup and view all the answers

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

<p>Combine solutions from the quadratic polynomial (C)</p> Signup and view all the answers

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

<p>The concept of limits (A)</p> Signup and view all the answers

What is the purpose of differential calculus in optimization problems?

<p>To find both the maximum and minimum values of a function (B)</p> Signup and view all the answers

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

<p>A straight line with a hole at x = -6 (A)</p> 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)?

<p>Because x + 6 is only defined when x is not equal to -6 (A)</p> Signup and view all the answers

What is the main application of differential calculus?

<p>All of the above (D)</p> Signup and view all the answers

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

<p>It is the foundation of calculus, allowing us to study rates of change and accumulation (D)</p> Signup and view all the answers

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

<p>To determine the rate of change of the function at that point (D)</p> Signup and view all the answers

What does the second derivative of a function indicate?

<p>The change in gradient of the original function (B)</p> 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?

<p>Their product is -1 (D)</p> 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?

<p>The graph falls to the right and rises to the left (C)</p> Signup and view all the answers

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

<p>By solving f(x) = 0 (B)</p> Signup and view all the answers

What is a local minimum of a cubic function?

<p>A point where the function changes from decreasing to increasing (A)</p> Signup and view all the answers

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

<p>To classify the stationary points as local maximum or local minimum (A)</p> Signup and view all the answers

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

<p>By using the point-slope form of the equation of a straight line (A)</p> Signup and view all the answers

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

<p>f''(x) (B)</p> Signup and view all the answers

What is the purpose of differential calculus in optimization problems?

<p>To find the stationary points of a function (C)</p> Signup and view all the answers

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

<p>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. (C)</p> Signup and view all the answers

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

<p>f'(x), y', Dy, and Df(x) (C)</p> Signup and view all the answers

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

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

What is the derivative of a constant k?

<p>The derivative of a constant k is 0. (A)</p> Signup and view all the answers

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

<p>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). (D)</p> Signup and view all the answers

What is the use of the rules for differentiation?

<p>The rules for differentiation are used to find the derivative of a function quickly and easily. (B)</p> Signup and view all the answers

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

<p>The gradient of the tangent to a curve is the derivative of the function at a point. (A)</p> Signup and view all the answers

What is the equation of a tangent to a curve?

<p>The equation of a tangent to a curve is y = mx + b, where m is the gradient and b is the y-intercept. (D)</p> Signup and view all the answers

When should you use the rules for differentiation?

<p>Whenever possible, to make finding derivatives faster and easier. (C)</p> Signup and view all the answers

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

<p>f''(x) and D^2y (A)</p> Signup and view all the answers

What is the definition of a concave up curve?

<p>A curve that opens upwards (C)</p> Signup and view all the answers

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

<p>To find the points of inflection of a function (D)</p> Signup and view all the answers

What is the formula for the remainder in synthetic division?

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

What is the definition of a point of inflection?

<p>A point where the curve changes concavity (B)</p> Signup and view all the answers

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

<p>To find the factors of the polynomial (A)</p> Signup and view all the answers

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

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

What is the graphical representation of a cubic function?

<p>An S-shaped curve (D)</p> Signup and view all the answers

What is the purpose of differential calculus in optimization problems?

<p>To find the maximum or minimum of a function (A)</p> Signup and view all the answers

What is the formula for long division of polynomials?

<p>a(x) = b(x) * Q(x) + R(x) (D)</p> Signup and view all the answers

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

<p>Solve f'(x) = 0 (B)</p> Signup and view all the answers

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

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

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

<p>One degree less than p(x) (A)</p> Signup and view all the answers

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

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

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

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

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

<p>Identify a factor by trial and error (A)</p> Signup and view all the answers

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

<p>To find a factor (D)</p> Signup and view all the answers

What is the expression for the quadratic formula?

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

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

<p>A factor corresponds to a root (C)</p> 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>p(x) = (cx - d) * Q(x) (B)</p> Signup and view all the answers

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

<p>Combine solutions (B)</p> Signup and view all the answers

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

<p>The function has a limit as x approaches -6 (A)</p> Signup and view all the answers

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

<p>The concept of limits (B)</p> Signup and view all the answers

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

<p>Because the denominator is zero at x = -6 (C)</p> 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?

<p>It illustrates the concept of limits (B)</p> Signup and view all the answers

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

<p>The function illustrates the concept of limits, which is fundamental to differential calculus (B)</p> Signup and view all the answers

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

<p>The concept of limits (B)</p> Signup and view all the answers

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

<p>-8 (B)</p> Signup and view all the answers

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

<p>2x + h (D)</p> Signup and view all the answers

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

<p>All of the above (D)</p> Signup and view all the answers

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

<p>The derivative of the sum is the sum of the derivatives (D)</p> Signup and view all the answers

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

<p>All of the above (D)</p> Signup and view all the answers

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

<p>The gradient of the tangent to the curve at the point (B)</p> 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)?

<p>All of the above (D)</p> Signup and view all the answers

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

<p>The derivative is the constant times the derivative of the function (D)</p> Signup and view all the answers

What is the purpose of using the rules for differentiation?

<p>To find the derivative of a function using shortcuts (A)</p> Signup and view all the answers

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

<p>nx^(n-1) (B)</p> 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?

<p>Their product is -1 (D)</p> Signup and view all the answers

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

<p>To determine the shape and orientation of the curve (B)</p> 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?

<p>The graph falls to the right and rises to the left (D)</p> Signup and view all the answers

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

<p>f''(x) (C)</p> Signup and view all the answers

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

<p>To find the maximum or minimum values of the function (A)</p> Signup and view all the answers

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

<p>y - y1 = m(x - x1) (A)</p> 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?

<p>The gradient is independent of the y-coordinate (B)</p> Signup and view all the answers

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

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

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

<p>To describe rates of change (B)</p> Signup and view all the answers

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

<p>A point where the function changes from increasing to decreasing (B)</p> 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?

<p>The function is concave up. (A)</p> 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?

<p>To determine the concavity of the function. (A)</p> 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>p(d/c) (A)</p> 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>p(d/c) = 0 (B)</p> 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>p(x) / (x - d/c) (C)</p> 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?

<p>The polynomial has a turning point at x = k. (D)</p> Signup and view all the answers

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

<p>To sketch the graph of the polynomial. (D)</p> 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?

<p>The first derivative is non-zero, and the second derivative is zero. (A)</p> Signup and view all the answers

What is the purpose of differential calculus in optimization problems?

<p>To find the stationary points of a function. (C)</p> Signup and view all the answers

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

<p>To find the instantaneous rate of change of a function. (C)</p> 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>p(x) = (cx - d)Q(x) + R (B)</p> Signup and view all the answers

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

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

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

<p>cx - d is a factor of p(x) (A)</p> 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?

<p>One degree less than p(x) (A)</p> Signup and view all the answers

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

<p>Find a factor by trial and error (C)</p> 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>cx - d is a factor of p(x) (C)</p> Signup and view all the answers

What is the expression for the quadratic formula?

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

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

<p>The roots correspond to the factors (C)</p> 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?

<p>Zero (D)</p> Signup and view all the answers

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

<p>To factorize the polynomial (C)</p> Signup and view all the answers

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