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</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</p> Signup and view all the answers

    What is the purpose of differential calculus?

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

    What is the derivative of a constant k?

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

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

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

    What is the definition of a point of inflection?

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

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

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

    What is the formula for the remainder in synthetic division?

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

    What is the definition of concave up?

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

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

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

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

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

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

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

    What is the expression for the quadratic formula?

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

    What is the main application of differential calculus?

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

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

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

    What is the derivative of a constant k?

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

    What is the definition of a concave up curve?

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

    What is the formula for the remainder in synthetic division?

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

    What is the definition of a point of inflection?

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

    What is the graphical representation of a cubic function?

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

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

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

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

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

    What is the expression for the quadratic formula?

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

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

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

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

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

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

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

    What is the expression for the quadratic formula?

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

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

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

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