Calculus 1 Comprehensive Final Exam
Review
Course Reference & Review Sheet | Target Exam Date:
Date | Instructor:
Person ​
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1. Limits and Continuity
Essential Limit Laws
For limits where $\lim_{x \to a} f(x) = L$ and $\lim_{x \to a} g(x) = M$:
●​ Sum / Difference Rule: $\lim_{x \to a} [f(x) \pm g(x)] = L \pm M$
●​ Product Rule: $\lim_{x \to a} [f(x) \cdot g(x)] = L \cdot M$
●​ Quotient Rule: $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{L}{M}$, provided $M
\neq 0$
Definition of Continuity
A function $f(x)$ is continuous at a point $x = c$ if and only if three conditions
are satisfied:
1.​ $f(c)$ is defined (i.e., $c$ is in the domain of $f$).
2.​ $\lim_{x \to c} f(x)$ exists.
3.​ $\lim_{x \to c} f(x) = f(c)$.
Squeeze Theorem
If $g(x) \le f(x) \le h(x)$ for all $x$ near $a$ (except possibly at $a$), and:​
$$\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$$​
Then:​
$$\lim_{x \to a} f(x) = L$$
2. Derivatives & Differentiation Techniques
Limit Definition of the Derivative
$$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$
Standard Differentiation Rules Reference Table
Rule Name
Function $f(x)$
Derivative $f'(x)$
Power Rule
$x^n$
$n x^{n-1}$
Product Rule
$u(x) \cdot v(x)$
$u'(x)v(x) + u(x)v'(x)$
Quotient Rule
$\frac{u(x)}{v(x)}$
Chain Rule
$f(g(x))$
$f'(g(x)) \cdot g'(x)$
Exponential
$e^{k x}$
$k e^{k x}$
Natural Logarithm
$\ln(x)$
$\frac{1}{x}$
Sine / Cosine
$\sin(x)$ / $\cos(x)$
$\cos(x)$ / $-\sin(x)$
Tangent
$\tan(x)$
$\sec^2(x)$
Inverse Tangent
$\arctan(x)$
$\frac{1}{1 + x^2}$
$\frac{u'(x)v(x) u(x)v'(x)}{[v(x)]^2}$
Implicit Differentiation Workflow
1.​ Differentiate both sides of the equation with respect to $x$.
2.​ Apply the Chain Rule whenever differentiating a term containing $y$
(yielding $\frac{dy}{dx}$).
3.​ Collect all terms containing $\frac{dy}{dx}$ on one side of the equation.
4.​ Factor out $\frac{dy}{dx}$ and solve algebraically.
3. Applications of Differentiation
Mean Value Theorem (MVT)
If $f(x)$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then there exists
at least one point $c \in (a, b)$ such that:​
$$f'(c) = \frac{f(b) - f(a)}{b - a}$$
Curve Sketching & Analysis Checklist
●​ First Derivative Test ($f'(x)$):
○​ Find critical numbers where $f'(x) = 0$ or $f'(x)$ is undefined.
○​ $f'(x) > 0 \implies$ Function is increasing.
○​ $f'(x) < 0 \implies$ Function is decreasing.
○​ Sign change $(+) \to (-) \implies$ Local maximum.
○​ Sign change $(-) \to (+) \implies$ Local minimum.
●​ Second Derivative Test ($f''(x)$):
○​ $f''(x) > 0 \implies$ Concave up ($\cup$).
○​ $f''(x) < 0 \implies$ Concave down ($\cap$).
○​ Inflection points occur where $f''(x)$ changes sign and $f(x)$ is
continuous.
L'Hôpital's Rule
If a limit yields an indeterminate form of $\frac{0}{0}$ or $\frac{\pm \infty}{\pm
\infty}$:​
$$\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$$​
Note: Always verify that the conditions hold before applying this rule. Do not use
the quotient rule here.
4. Integration & The Fundamental Theorem of
Calculus
Fundamental Theorem of Calculus (FTC)
●​ Part 1 (Evaluation):​
If $F(x)$ is an antiderivative of $f(x)$, then:​
$$\int_{a}^{b} f(x) , dx = F(b) - F(a)$$
●​ Part 2 (Derivative of an Integral):​
$$\frac{d}{dx} \left[ \int_{a}^{x} f(t) , dt \right] = f(x)$$​
With Chain Rule extension:​
$$\frac{d}{dx} \left[ \int_{a}^{g(x)} f(t) , dt \right] = f(g(x)) \cdot g'(x)$$
Integration by Substitution ($u$-Substitution)
Used to invert the Chain Rule for integrals of the form $\int f(g(x)) g'(x) , dx$:
1.​ Select $u = g(x)$ (typically the inside function).
2.​ Compute $du = g'(x) , dx \implies dx = \frac{du}{g'(x)}$.
3.​ Substitute to rewrite the entire integral in terms of $u$.
4.​ Evaluate the integral, then substitute $g(x)$ back in for $u$ (or update
integration limits for definite integrals).
5. Common Exam Pitfalls & High-Yield Reminders
●​ Don't Forget $+ C$: Always append the constant of integration $+ C$ to
indefinite integrals.
●​ Chain Rule Oversight: When taking derivatives of composite functions (like
$\sin(x^2)$), ensure you multiply by the derivative of the inner function ($2x
\cos(x^2)$).
●​ Domain Restrictions: Always check for division by zero, negative values
inside logarithms, or negative values under even roots when identifying
critical points or vertical asymptotes.
●​ Definite Integral Substitution: When applying $u$-substitution to definite
integrals, either change the limits of integration to match $u$ or substitute
back to $x$ before applying original limits.