Gr12 Mathematics: Ch 5.2 Differentiation from first Principles
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Questions and Answers

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

  • f'(x) = [f(x + h) + f(x)]/h
  • f'(x) = [f(x + h) - f(x)]/h
  • f'(x) = lim(h → ∞) [f(x + h) - f(x)]/h
  • f'(x) = lim(h → 0) [f(x + h) - f(x)]/h (correct)

    • What is the purpose of differentiation from first principles?

  • To find the maximum value of a function
  • To determine the gradient of a tangent to a curve (correct)
  • To find the intersection points of two curves
  • To find the area under a curve

    • What is the meaning of the notation dy/dx?

  • The quotient of y and x
  • y differentiated with respect to x (correct)
  • The product of y and x
  • x differentiated with respect to y

    • What is the name given to the symbols D and d/dx?

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

    What is the formula for the gradient of a tangent to a curve at a point x = a?

    <p>lim(h → 0) [f(a + h) - f(a)]/h</p> Signup and view all the answers

    What is the common notation used to denote the derivative of a function f(x)?

    <p>f'(x) = y' = dy/dx</p> Signup and view all the answers

    Why is the notation dy/dx not considered a fraction?

    <p>Because it represents a limit</p> Signup and view all the answers

    What is the term used to describe the process of finding the derivative of a function?

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

    What is the purpose of the formula ∂Gradient at a point = lim(h → 0) [f(a + h) - f(a)]/h?

    <p>To determine the gradient of the tangent to a curve at a point</p> Signup and view all the answers

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

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

    What does the notation Dy denote?

    <p>The derivative of y with respect to x</p> Signup and view all the answers

    What is the term used to describe the process of finding the derivative of a function?

    <p>Differentiation from first principles</p> Signup and view all the answers

    What is the differential operator denoted by?

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

    What does the notation dp/dx mean?

    <p>p differentiated with respect to x</p> Signup and view all the answers

    What is the importance of being consistent in using notations for derivatives?

    <p>To avoid confusion</p> Signup and view all the answers

    What is the relationship between the derivative of a function and the gradient of the tangent to the graph?

    <p>The derivative is the gradient of the tangent to the graph</p> Signup and view all the answers

    What is the primary purpose of using the formula for the derivative to determine the gradient of a tangent to a curve?

    <p>To find the expression for the gradient of the graph at any point.</p> Signup and view all the answers

    If we have a function ( f(x) = x^2 ), what is the significance of the notation ( f'(x) )?

    <p>It represents the rate of change of the function with respect to ( x ).</p> Signup and view all the answers

    Which of the following notations is NOT a valid way to denote the derivative of a function ( f(x) )?

    <p>( \int f(x) dx )</p> Signup and view all the answers

    What is the relationship between the derivative of a function and the gradient of the tangent to the graph?

    <p>The derivative of a function is equal to the gradient of the tangent to the graph.</p> Signup and view all the answers

    What is the purpose of the differential operator ( D ) in the notation ( Df(x) )?

    <p>To indicate the differentiation of the function.</p> Signup and view all the answers

    If we have a function ( f(x) = x^2 ) and we want to find the derivative of the function with respect to ( x ), what is the correct notation to use?

    <p>( rac{df}{dx} )</p> Signup and view all the answers

    What is the significance of the notation ( rac{dy}{dx} ) in the context of differentiation?

    <p>It represents the derivative of the function with respect to ( x ).</p> Signup and view all the answers

    Why is it important to be consistent in using notations for derivatives?

    <p>To avoid confusion and ensure clarity in mathematical expressions.</p> Signup and view all the answers

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