1.State the mathematical condition for a function $f(x)$ to be continuous at a point $x = c$.
2.True or False: Every polynomial function is continuous everywhere on the real line.
3.Find the Left Hand Limit (LHL) and Right Hand Limit (RHL) of $f(x) = 2x+3$ at $x = 1$.
4.Is the constant function $f(x) = k$ continuous at $x = 0$?
5.If $f(x)$ and $g(x)$ are continuous at $x=c$, then what can you say about the continuity of $(f+g)(x)$ at $x=c$?
6.At what point is the function $f(x) = \frac{1}{x-3}$ discontinuous?
7.True or False: The modulus function $f(x) = |x|$ is continuous at $x=0$.
8.Is the sine function, $f(x) = \sin x$, continuous everywhere?
9.Identify the points of discontinuity for the Greatest Integer Function $f(x) = [x]$.
10.If $g(x)$ is continuous at $c$ and $f(x)$ is continuous at $g(c)$, is the composition $f(g(x))$ continuous at $c$?
11.State the relationship between continuity and differentiability. (Does differentiability imply continuity?)
12.True or False: A continuous function is always differentiable.
13.Is the function $f(x) = |x|$ differentiable at $x = 0$?
14.What is the derivative of $x^n$ with respect to $x$ (Power Rule)?
15.What is the derivative of any constant value $C$?
16.Write the formula for the Product Rule: $\frac{d}{dx}[u(x)v(x)]$.
17.Write the formula for the Quotient Rule: $\frac{d}{dx}\left[\frac{u(x)}{v(x)}\right]$.
18.Find the derivative of $y = x^3 + 5x - 7$.
19.Find the derivative of $y = x \cdot \sin x$.
20.Find the derivative of $y = \frac{x}{\cos x}$.
21.State the Chain Rule for differentiating a composite function $y = f(g(x))$.
22.Find the derivative of $y = \sin(2x)$.
23.Find the derivative of $y = \cos(x^2)$.
24.Find the derivative of $y = (ax+b)^n$.
25.Find the derivative of $y = \sqrt{x^2+1}$.
26.What does it mean for an equation to be given implicitly?
27.If $x + y = \pi$, find $\frac{dy}{dx}$.
28.If $y - \sin y = x$, differentiate both sides with respect to $x$ to find $\frac{dy}{dx}$.
29.Differentiate $y^2$ with respect to $x$.
30.If $xy = c$, find $\frac{dy}{dx}$.
31.Write the formula for $\frac{d}{dx}(\sin^{-1}x)$.
32.Write the formula for $\frac{d}{dx}(\cos^{-1}x)$.
33.Write the formula for $\frac{d}{dx}(\tan^{-1}x)$.
34.Write the formula for $\frac{d}{dx}(\sec^{-1}x)$.
35.Write the formula for $\frac{d}{dx}(\cot^{-1}x)$.
36.Find the derivative of $y = \sin^{-1}(2x)$.
37.Find the derivative of $y = \tan^{-1}\left(\frac{x}{a}\right)$.
38.Find the derivative of $y = \cos^{-1}(\sin x)$.
39.What is the best trigonometric substitution for simplifying expressions with $\sqrt{1-x^2}$?
40.What is the best trigonometric substitution for simplifying expressions with $1+x^2$?
41.What is the derivative of the natural exponential function $e^x$?
42.What is the derivative of $a^x$ (where $a > 0, a \neq 1$)?
43.What is the derivative of the natural logarithm $\log_e x$ (or $\ln x$)?
44.Find the derivative of $y = e^{3x}$.
45.Find the derivative of $y = \log(\sin x)$.
46.When is the method of "Logarithmic Differentiation" most useful?
47.What is the first step in differentiating a function like $y = x^x$?
48.Using logarithmic properties, expand $\log(u \cdot v \cdot w)$.
49.Find $\frac{d}{dx}(x^x)$. (Standard Result)
50.Simplify and find the derivative of $y = e^{\log_e x}$.
51.If $x = f(t)$ and $y = g(t)$, state the formula to find $\frac{dy}{dx}$.
52.If $x = 2at$ and $y = at^2$, find $\frac{dx}{dt}$ and $\frac{dy}{dt}$.
53.Using the previous question's functions, find $\frac{dy}{dx}$.
54.If $x = r\cos\theta$ and $y = r\sin\theta$, find $\frac{dy}{dx}$.
55.What is the symbol and definition for the Second Order Derivative?
56.Find the second order derivative ($\frac{d^2y}{dx^2}$) of $y = x^3$.
57.Find the second order derivative of $y = \sin x$.
58.Find the second order derivative of $y = e^{2x}$.
59.True or False: $\frac{d^2y}{dx^2}$ is exactly the same as $\left(\frac{dy}{dx}\right)^2$.
60.What is the second derivative of a linear function $y = mx + c$?
61.What are the three required conditions for Rolle's Theorem to be applicable on a function $f(x)$ in interval $[a, b]$?
62.If the conditions of Rolle's Theorem are satisfied, what is the conclusion regarding $f'(c)$ for some $c \in (a, b)$?
63.What are the two conditions for Lagrange's Mean Value Theorem (LMVT) to be applicable?
64.State the formula representing the conclusion of LMVT.
65.Geometrically, what does Rolle's Theorem imply about the tangent at point $c$?
