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What concept does Zeno's paradox of Achilles and the tortoise primarily illustrate?

For the function $y = \frac{(x + 6)(x - 2)}{x + 6}$, what value does $y$ approach as $x$ approaches -6?

What is the correct notation for the limit of the function $f(x) = \frac{(x + 6)(x - 2)}{x + 6}$ as $x$ approaches -6?

Using differentiation from first principles, which expression defines the derivative $f'(x)$ of a function $f(x)$?

Which of the following notations does NOT represent the derivative of $y = f(x)$?

According to the rules of differentiation, what is the derivative of a constant $k$ with respect to $x$?

What is the derivative of $5x^3$ with respect to $x$?

What is the derivative of the sum of two functions, $f(x)$ and $g(x)$, with respect to $x$?

When should you use differentiation from first principles to find the derivative of a function?

What is the relationship between the gradient of a curve at a point and the gradient of the tangent to the curve at that same point?

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

If the gradient of the tangent to a curve at a point is 2, what is the gradient of the normal to the curve at that point?

What does the second derivative of a function indicate?

If $f''(x) > 0$, what does this indicate about the concavity of the graph of $f(x)$?

Which of the following is NOT a common use of the derivative?

What is the general form of a cubic function?

For a cubic function $f(x) = ax^3 + bx^2 + cx + d$, how does the sign of $a$ affect the shape and orientation of the graph?

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

What is a stationary point on a curve?

How are local maximum and local minimum points defined for cubic functions?

What is a point of inflection?

In the context of sketching cubic graphs, what does concavity indicate?

What is the first step in the general method for sketching cubic graphs?

What is the purpose of using differential calculus in optimization problems?

How does the derivative relate to the rate of change?

According to the division rule for polynomials, what is the correct expression when dividing polynomial $a(x)$ by $b(x)$?

Which of the following is NOT a method for factorizing cubic polynomials?

In synthetic division, what sign should you use for the root of the divisor polynomial?

Given the cubic polynomial $f(x) = ax^3 + bx^2 + cx + d$, what is the first step in drawing its graph?

For a cubic polynomial $ax^3 + bx^2 + cx + d = 0$, if $f(k) = 0$, what does this imply according to the Factor and Remainder Theorem?

What does the Remainder Theorem state?

According to the Remainder Theorem, if you divide a polynomial $p(x)$ by $cx - d$, what is the remainder $R$?

In the context of the Remainder Theorem, what is the degree of the quotient $Q(x)$ when dividing a polynomial $p(x)$ by a linear polynomial $cx - d$?

What does the Factor Theorem state?

If $cx - d$ is a factor of polynomial $p(x)$, how can $p(x)$ be expressed?

In solving cubic equations, what role does the Quadratic Formula play?

What is the first step in solving a cubic equation using factorization methods?

After finding a factor of a cubic polynomial, what is the next step in solving the cubic equation?

Given the function $f(x) = 3x^2 - 2x + 1$, what is the expression for $f'(x)$ using the rules of differentiation?

If the tangent to a curve at the point $x = 2$ has a gradient of -3, what is the gradient of the normal to the curve at that point?

Given $f(x) = x^3 - 6x^2 + 5$, find the interval(s) where $f''(x) > 0$.

Consider a cubic function $f(x) = ax^3 + bx^2 + cx + d$ with $a < 0$. Which statement accurately describes the end behavior of the graph?

For the cubic function $f(x) = x^3 - 3x^2 + 2$, what are the x-coordinates of the stationary points?

A curve is defined by the equation $y = x^3 - 6x^2$. Determine the interval(s) where the curve is concave up.

To optimize variable $A$, which is expressed in terms of variable $x$, using differential calculus, what is the condition to find the minimum value of $A$?

Given the polynomial division $a(x) = (x + 2)(x^2 - x + 3) + R(x)$, what is the degree of the remainder $R(x)$?

When using synthetic division to divide the polynomial $x^3 - 4x^2 + x + 6$ by $(x - 2)$, what value should be used for the root of the divisor?

Given that $p(x) = (x - 3)Q(x) + 5$, what is the value of $p(3)$ according to the Remainder Theorem?

In the context of limits, what does it mean for $\lim_{x \to a} f(x) = L$?

For a function $f(x)$ that is not defined at $x = c$, but $\lim_{x \to c} f(x)$ exists, what can be said about the graph of $f(x)$ at $x = c$?

Given $f(x) = x^2$, what expression represents the derivative $f'(x)$ obtained using differentiation from first principles?

Which of the following expressions represents the derivative of $f(x)$ with respect to $x$?

If $f(x) = kx$, where $k$ is a constant, what represents $f'(x)$?

What is the derivative of $f(x) = x^4 + 2x^2 - 5x + 3$ with respect to $x$?

Given $f(x) = u(x) - v(x)$, where $u(x)$ and $v(x)$ are differentiable functions, what is the derivative of $f(x)$ with respect to $x$?

Under what condition is differentiation from first principles absolutely necessary to find the derivative?

What is the gradient of the tangent to the curve $y = f(x)$ at the point where $x = a$?

To find the equation of a tangent to a curve at a given point, what must you first determine?

If the gradient of the tangent to a curve at a certain point is $m$, what is the gradient of the normal to the curve at that point?

What does the second derivative, $f''(x)$, reveal about the function $f(x)$?

If $f''(x) < 0$ over an interval, what does this indicate about the shape of the graph of $f(x)$ in that interval?

Which of these problems cannot be solved using derivatives?

What is the general equation form of a cubic function?

For a cubic function $f(x) = ax^3 + bx^2 + cx + d$, how does the sign of 'a' influence the graph's end behavior?

How is the y-intercept of a cubic function $f(x) = ax^3 + bx^2 + cx + d$ determined?

Which of the following is true about a stationary point on a curve?

For a cubic function, what distinguishes a local maximum from a local minimum?

What condition defines a point of inflection on a curve?

How does concavity relate to the gradient of a curve?

When initially sketching a cubic graph, what is the significance of determining the sign of the leading coefficient 'a'?

What fundamental principle underlies using differential calculus in optimization problems?

How does the derivative of a function concretely relate to its rate of change?

In polynomial division, if $a(x)$ is divided by $b(x)$ resulting in a quotient $Q(x)$ and a remainder $R(x)$, which equation correctly represents this division?

Besides long division, what is another method for factorizing cubic polynomials?

When using synthetic division, what is the significance of the 'root of the divisor polynomial'?

What initial step is crucial when graphing a cubic polynomial function $f(x) = ax^3 + bx^2 + cx + d$?

According to the Factor and Remainder Theorem, what does $f(k) = 0$ imply for the cubic polynomial $f(x)$?

What key concept does the Remainder Theorem provide for polynomial algebra?

According to the Remainder Theorem, if you divide a polynomial $p(x)$ by $cx - d$, how do you find the remainder, $R$?

In the context of the Remainder Theorem, when dividing a polynomial $p(x)$ by a linear polynomial $cx - d$, what is the nature of the remainder R?

What does the Factor Theorem fundamentally assert?

If $cx - d$ is a factor of the polynomial $p(x)$, how can $p(x)$ be expressed according to the Factor Theorem?

What role does the Quadratic Formula play in solving cubic equations?

In solving cubic equations via factorization, what is a critical initial step?

After successfully identifying a factor of a cubic polynomial, what is the subsequent step in solving the cubic equation?

Given a function $f(x)$, which of the following expressions represents a situation where simply substituting a value into the function will NOT give you the limit?

Assume $f(x)$ and $g(x)$ are differentiable functions. Characterize what happens if you know that at point $x=a$, $f'(a) = g'(a)$?

You're tasked to rigorously prove that $\lim_{x \to c} x^2 = c^2$ directly from the $\epsilon$-$\delta$ definition of a limit. Which approach would be suitable?

If the limit of $f(x)$ as $x$ approaches $a$ exists and is equal to $L$, which of the following statements is always true?

Which of the following is true about the graph of a function $f(x)$ at $x = c$, if $f(c)$ is undefined but $\lim_{x \to c} f(x) = L$?

Given the function $f(x) = x^3$, determine $f'(x)$ using differentiation from first principles.

Given a differentiable function $f(x)$, which of the following expressions is NOT a valid representation of its derivative?

If $f(x) = 7x$, what is $f'(x)$?

Determine the derivative of the polynomial function $f(x) = 6x^5 - 4x^3 + 2x - 9$ with respect to $x$.

Given $f(x) = u(x) + v(x)$, where both $u(x)$ and $v(x)$ are differentiable functions, find the derivative of $f(x)$.

Under which condition is it essential to use differentiation from first principles to find the derivative?

Given a function $f(x)$, the tangent line at $x = a$ is horizontal. What does this imply about the value of $f'(a)$?

Suppose a function $f(x)$ is differentiable on an interval $(a, b)$, and for some $c$ in $(a, b)$, $f'(c) = 0$ and $f''(c) = 0$. Which of the following is definitively true at $x = c$?