Practice Questions

Critique the following evaluation of a limit and justify why it is incorrect: $\lim_{x \to 3} \frac{x^2 - 9}{x - 3} = \frac{3^2 - 9}{3 - 3} = \frac{0}{0} = 1$.

Explain the condition required for the limit of a function $f(x)$ to exist at a point $x=a$.

State the formula for the derivative of a function $f(x)$ at a point $x=a$ using the first principle.

Calculate the derivative of the function $f(x) = 7x^3 - 4x^2 + 2x - 9$.

Calculate the limit: $\lim_{x \to 4} \frac{x^2 - x - 12}{x - 4}$

Define the Right Hand Limit of a function $f(x)$ at a point $x=a$.

What is the derivative of a constant function $f(x) = c$, where $c$ is a real number?

Identify the standard derivatives for the following functions: (i) $f(x) = x^n$ (for any positive integer $n$) (ii) $f(x) = \sin x$ (iii) $f(x) = \cos x$

State the product rule for derivatives. If $u=f(x)$ and $v=g(x)$, what is the formula for the derivative of their product $uv$?

Calculate the limit: $\lim_{x \to 2} (3x^2 - x + 5)$

Recall the standard limit formula for $\lim_{x \rightarrow a} \frac{x^n - a^n}{x - a}$.

State the Sandwich Theorem for limits.

State the quotient rule for derivatives. If $u=f(x)$ and $v=g(x)$, what is the formula for the derivative of their quotient $u/v$?

Describe the relationship between the derivative of a function at a point and the tangent to the curve at that point.

Calculate the derivative of $f(x) = x^3 \cos x$ using the product rule.

Summarize the intuitive idea of a derivative as explained through the example of a body's velocity. Explain what average velocity and instantaneous velocity represent in this context.

Calculate the derivative of $f(x) = \frac{2x - 3}{x^2 + 1}$ using the quotient rule.

List and explain the four main rules from the algebra of limits (Theorem 1). For each rule, let $\lim_{x \rightarrow a} f(x)$ and $\lim_{x \rightarrow a} g(x)$ exist.

Solve for the limit: $\lim_{x \to 2} \frac{x^5 - 32}{x^2 - 4}$

Calculate the derivative of $f(x) = \frac{1}{2x+3}$ from first principles.

Evaluate the limit: $\lim_{x \to 2} \left( \frac{1}{x-2} - \frac{4}{x^2-4} \right)$

Calculate the derivative of $f(x) = \sec x - 10 \tan x$.

Formulate a single, non-piecewise rational function $f(x)$ that is defined for all real numbers except $x=5$, and whose limit as $x \to 5$ is 10.

Justify, by evaluating the left-hand and right-hand derivatives, why the function $f(x) = |x+1|$ is not differentiable at $x=-1$.

Formulate a rational function $f(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials, such that $\lim_{x \to 1} f(x)$ initially presents as the indeterminate form $\frac{0}{0}$ but ultimately evaluates to 6.

Justify whether the function $f(x) = \begin{cases} x^2+1, & x \le 1 \ 3x-1, & x > 1 \end{cases}$ is differentiable at $x=1$.

Evaluate the limit: $\lim_{x \to 0} \frac{\sin(7x)}{3x}$

List the two important trigonometric limits involving $\sin x$ and $\cos x$ as $x \rightarrow 0$.

Evaluate the derivative of $y = \frac{x \sin x}{1 + x}$ and justify each major step by naming the differentiation rule used.

Given that $\lim_{x \to 1} \frac{ax^2 + bx - 4}{x - 1} = 8$, formulate a system of equations to find the constants $a$ and $b$. Justify your reasoning and solve for the constants.

Evaluate the limit $\lim_{x \to \frac{\pi}{4}} \frac{1 - \tan x}{\cos(2x)}$ and justify your method.

Formulate a proof from first principles for the Product Rule of differentiation. That is, if $k(x) = f(x)g(x)$, prove that $k'(x) = f'(x)g(x) + f(x)g'(x)$, assuming $f$ and $g$ are differentiable.

Analyze the function $f(x) = \begin{cases} x^2 - 1, & x < 1 \ a, & x = 1 \ b - x^3, & x > 1 \end{cases}$ If $\lim_{x \to 1} f(x) = f(1)$, calculate the values of the constants $a$ and $b$.

Derive the formula for the derivative of $f(x) = \sin(x^2)$ from first principles. Justify all key steps in your derivation, including the use of trigonometric identities and standard limits.

Explain the concept of a rational function and describe how to find the limit of a rational function $f(x) = \frac{g(x)}{h(x)}$ as $x \rightarrow a$, when $h(a) \neq 0$.

List and describe the four main rules from the algebra of derivatives (Theorem 5). For each rule, let $u=f(x)$ and $v=g(x)$.

Calculate the derivative of $f(x) = x^2 - 4x$ from first principles.

Calculate the derivative of the function $f(x) = \frac{x \sin x}{1 + \cos x}$.

Solve for the limit: $\lim_{x \to \pi} \frac{x - \pi}{\sin x}$

Evaluate the limit: $\lim_{x \to 0} \frac{x \tan x}{1 - \cos(2x)}$

Justify the steps required to evaluate the limit $\lim_{x \to 0} \frac{1 - \cos(4x)}{x^2}$.

Design a polynomial function $f(x)$ of the lowest possible degree that satisfies the following two conditions: $\lim_{x \to 0} \frac{f(x)}{x^2} = 5$ and $\lim_{x \to 0} \frac{f(x)}{x^3}$ does not exist.

Propose a non-constant function $f(x)$ such that the derivative of the product $x^2 f(x)$ is $6x^2$.

Derive the formula for the derivative of $f(x) = \sec x$ from first principles.

Critique the following statement and justify your conclusion with a counterexample: 'If $\lim_{x \to a} f(x)$ and $\lim_{x \to a} g(x)$ both do not exist, then $\lim_{x \to a} [f(x) \cdot g(x)]$ also cannot exist.'