MA 161 Final Exam Study Guide: Core Calculus Concepts and Applications

Functions

Domain and Range

The domain of a function is the set of all possible input values (x-values) for which the function is defined. The range is the set of all possible output values (y-values) the function can produce.

• Key Point: To find the domain, identify values that cause division by zero, negative square roots, or undefined expressions.

• Key Point: The range depends on the function's formula and its domain.

• Example: For $f(x) = \sqrt{x-2}$, the domain is $x \geq 2$ and the range is $y \geq 0$.

Graphing and Transformations

Graph transformations include shifts, stretches, compressions, and reflections.

• Key Point: $f(x+a)$ shifts the graph left by $a$ units; $f(x)-a$ shifts it down by $a$ units.

• Key Point: $af(x)$ stretches vertically by $a$; $f(ax)$ compresses horizontally by $a$.

• Example: $g(x) = 2f(x-3) + 1$ is $f(x)$ stretched vertically by 2, shifted right by 3, and up by 1.

Inverses, Exponentials, and Logarithms

Basic Properties and Laws

Exponential and logarithmic functions are inverses of each other. Their properties are essential for solving equations and simplifying expressions.

• Key Point: $a^{x+y} = a^x \cdot a^y$; $\ln(ab) = \ln a + \ln b$

• Key Point: $\log_b(x) = \frac{\ln x}{\ln b}$

• Example: $\ln(e^x) = x$

Graphs and Interpretations

Exponential functions grow rapidly; logarithmic functions grow slowly. Their graphs reflect these behaviors.

• Key Point: $y = e^x$ passes through $(0,1)$ and increases as $x$ increases.

• Key Point: $y = \ln x$ is undefined for $x \leq 0$ and increases slowly for large $x$.

Trigonometric Functions and Their Inverses

Unit Circle and Basic Graphs

The unit circle helps define sine, cosine, and tangent functions. Their graphs are periodic and have specific amplitudes and periods.

• Key Point: $\sin(\theta)$ and $\cos(\theta)$ oscillate between -1 and 1.

• Key Point: The period of $\sin$ and $\cos$ is $2\pi$.

• Example: $\sin(\pi/2) = 1$

Evaluating Expressions Using Right Triangles

Trigonometric values can be found using right triangle ratios.

• Key Point: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$

• Key Point: $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$

Limits

One-Sided Limits

One-sided limits consider the behavior of a function as $x$ approaches a value from the left ($-$) or right ($+$).

• Key Point: $\lim_{x \to a^-} f(x)$ and $\lim_{x \to a^+} f(x)$

• Example: For $f(x) = \frac{1}{x}$, $\lim_{x \to 0^+} f(x) = +\infty$

Limits from Graphs

Limits can be estimated by observing the graph near the point of interest.

• Key Point: If the left and right limits are equal, the limit exists.

Algebraic Techniques

Algebraic methods help evaluate limits that are indeterminate.

• Key Point: Factor and cancel common terms.

• Key Point: Multiply by the conjugate to simplify square roots.

• Example: $\lim_{x \to 2} \frac{x^2-4}{x-2}$: Factor numerator to get $\frac{(x-2)(x+2)}{x-2}$, so limit is 4.

Squeeze Theorem

The Squeeze Theorem is used when a function is bounded between two others whose limits are known.

• Key Point: If $g(x) \leq f(x) \leq h(x)$ and $\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$, then $\lim_{x \to a} f(x) = L$.

Infinite Limits and Vertical Asymptotes

Infinite limits occur when a function grows without bound near a point, often indicating a vertical asymptote.

• Key Point: $\lim_{x \to a} f(x) = \infty$ means $f(x)$ increases without bound as $x$ approaches $a$.

• Key Point: Vertical asymptotes occur where the denominator is zero and the numerator is nonzero.

Limits at Infinity and Horizontal Asymptotes

Limits as $x$ approaches infinity describe the end behavior of functions.

• Key Point: Horizontal asymptotes are found by evaluating $\lim_{x \to \infty} f(x)$.

• Example: $\lim_{x \to \infty} \frac{1}{x} = 0$

Continuity

Types of Discontinuities

Discontinuities are classified as holes (removable), jumps, or infinite (vertical asymptotes).

• Key Point: A function is continuous at $x=a$ if $\lim_{x \to a} f(x) = f(a)$.

• Key Point: Piecewise functions may have discontinuities at boundaries.

Intervals of Continuity and Intermediate Value Theorem

The Intermediate Value Theorem states that if a function is continuous on $[a,b]$, it takes every value between $f(a)$ and $f(b)$.

• Key Point: Continuous functions on intervals have no breaks, jumps, or holes.

Introducing the Derivative

Tangent Lines and Instantaneous Rate of Change

The derivative at a point gives the slope of the tangent line and the instantaneous rate of change.

• Key Point: The derivative is defined as $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$

• Example: For $f(x) = x^2$, $f'(a) = 2a$

Derivative as a Function

The derivative function $f'(x)$ describes the rate of change at every point.

• Key Point: Differentiability implies continuity, but not vice versa.

• Key Point: Piecewise functions may not be differentiable at boundaries.

Rules of Differentiation

Basic Rules

Several rules simplify the process of finding derivatives.

• Key Point: Power Rule: $\frac{d}{dx} x^n = n x^{n-1}$

• Key Point: Constant Multiple Rule: $\frac{d}{dx} [c f(x)] = c f'(x)$

• Key Point: Sum Rule: $\frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x)$

• Key Point: $\frac{d}{dx} e^x = e^x$

• Key Point: Product Rule: $\frac{d}{dx} [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$

• Key Point: Quotient Rule: $\frac{d}{dx} \left[ \frac{f(x)}{g(x)} \right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$

Trig Rules

Derivatives of trigonometric functions are fundamental in calculus.

• Key Point: $\frac{d}{dx} \sin x = \cos x$

• Key Point: $\frac{d}{dx} \cos x = -\sin x$

• Key Point: $\frac{d}{dx} \tan x = \sec^2 x$

Chain Rule

The Chain Rule is used for differentiating composite functions.

• Key Point: $\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$

• Example: $\frac{d}{dx} \sin(x^2) = \cos(x^2) \cdot 2x$

Implicit Differentiation

Implicit differentiation is used when functions are not solved for $y$ explicitly.

• Key Point: Differentiate both sides with respect to $x$, treating $y$ as a function of $x$.

• Example: For $x^2 + y^2 = 1$, $2x + 2y \frac{dy}{dx} = 0 \Rightarrow \frac{dy}{dx} = -\frac{x}{y}$

Logarithmic Differentiation

Logarithmic differentiation is useful for functions with variable exponents.

• Key Point: Take $\ln$ of both sides, then differentiate.

• Example: $y = x^x \Rightarrow \ln y = x \ln x \Rightarrow \frac{1}{y} \frac{dy}{dx} = \ln x + 1 \Rightarrow \frac{dy}{dx} = x^x (\ln x + 1)$

Derivatives of Inverse Functions

The derivative of an inverse function can be found using the formula:

• Key Point: If $f$ and $f^{-1}$ are inverses, $\left[ f^{-1} \right]'(y) = \frac{1}{f'(x)}$ where $x = f^{-1}(y)$

Related Rates

Related rates problems involve differentiating with respect to time to relate changing quantities.

• Key Point: Use implicit differentiation with respect to $t$.

• Example: If $x^2 + y^2 = r^2$, $2x \frac{dx}{dt} + 2y \frac{dy}{dt} = 0$

Mean Value Theorem and Rolle’s Theorem

Hypotheses and Conclusions

These theorems guarantee the existence of certain points with specific derivative values.

• Key Point: Mean Value Theorem: If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then $\exists c \in (a,b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$

• Key Point: Rolle’s Theorem: If $f(a) = f(b)$, then $\exists c \in (a,b)$ such that $f'(c) = 0$

Critical Points and Extrema

Finding Critical Points

Critical points occur where $f'(x) = 0$ or $f'(x)$ is undefined.

• Key Point: Local maxima and minima are found at critical points or endpoints.

• Key Point: Use the first or second derivative test to classify extrema.

First Derivative Test

The first derivative test uses sign changes in $f'(x)$ to classify critical points.

• Key Point: If $f'(x)$ changes from positive to negative, there is a local maximum.

• Key Point: If $f'(x)$ changes from negative to positive, there is a local minimum.

Second Derivative Test and Concavity

The second derivative test uses $f''(x)$ to classify critical points and determine concavity.

• Key Point: If $f''(c) > 0$, $f$ is concave up at $c$ (local minimum).

• Key Point: If $f''(c) < 0$, $f$ is concave down at $c$ (local maximum).

• Key Point: Inflection points occur where $f''(x)$ changes sign.

Optimization Problems

Translating Word Problems

Optimization involves maximizing or minimizing a function subject to constraints.

• Key Point: Express the quantity to be optimized as a function of one variable.

• Key Point: Use derivatives to find critical points and test endpoints.

• Example: Minimize surface area for a fixed volume.

Linearization and Differentials

Linearization

Linearization approximates a function near a point using its tangent line.

• Key Point: $L(x) = f(a) + f'(a)(x - a)$

• Example: Approximate $\sqrt{4.1}$ using $f(x) = \sqrt{x}$ at $a = 4$.

Differentials

Differentials estimate small changes in a function.

• Key Point: $dy = f'(x) dx$

L'Hôpital's Rule

Applying L'Hôpital's Rule

L'Hôpital's Rule is used to evaluate limits of indeterminate forms.

• Key Point: If $\lim_{x \to a} \frac{f(x)}{g(x)}$ is $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$

• Key Point: Other indeterminate forms can be converted to quotients.

Antiderivatives

Indefinite Integrals

Antiderivatives are functions whose derivative is the given function.

• Key Point: $\int f(x) dx = F(x) + C$

• Key Point: The constant of integration $C$ accounts for all possible antiderivatives.

Approximating Areas Under Curves

Riemann Sums

Riemann sums approximate the area under a curve by summing areas of rectangles.

• Key Point: Left Riemann sum uses left endpoints; right Riemann sum uses right endpoints.

• Key Point: Area can also be calculated using geometric shapes.

Definite Integrals

Area Interpretation

The definite integral represents the net area under a curve between two points.

• Key Point: $\int_a^b f(x) dx$ is the area under $f(x)$ from $x=a$ to $x=b$.

• Key Point: The area function $A(x) = \int_a^x f(t) dt$

Fundamental Theorem of Calculus

Part 1 and Part 2

The Fundamental Theorem of Calculus connects differentiation and integration.

• Key Point: Part 1: If $F(x) = \int_a^x f(t) dt$, then $F'(x) = f(x)$

• Key Point: Part 2: $\int_a^b f(x) dx = F(b) - F(a)$ where $F$ is any antiderivative of $f$

• Key Point: Substitution can be used to apply Part 1.

Working with Integrals

Odd and Even Symmetry

Symmetry properties can simplify definite integrals.

• Key Point: If $f$ is even, $\int_{-a}^a f(x) dx = 2 \int_0^a f(x) dx$

• Key Point: If $f$ is odd, $\int_{-a}^a f(x) dx = 0$

Average Value of a Function

The average value of $f$ on $[a,b]$ is given by:

• Key Point: $\text{Average} = \frac{1}{b-a} \int_a^b f(x) dx$

Substitution Rule

U-Substitution

U-substitution simplifies integrals by changing variables.

• Key Point: Let $u = g(x)$, then $\int f(g(x)) g'(x) dx = \int f(u) du$

• Example: $\int 2x \cos(x^2) dx$; let $u = x^2$, $du = 2x dx$, so $\int \cos(u) du = \sin(u) + C$

Exponential Models

Exponential Growth and Decay

Exponential models describe processes that grow or decay at rates proportional to their current value.

• Key Point: $y(t) = y_0 e^{kt}$, where $k > 0$ for growth, $k < 0$ for decay

• Key Point: Half-life: $t_{1/2} = \frac{\ln 2}{|k|}$

• Key Point: Doubling time: $t_{2} = \frac{\ln 2}{k}$

Type	Formula	Interpretation
Growth	$y(t) = y_0 e^{kt}$	Population increases over time
Decay	$y(t) = y_0 e^{-kt}$	Quantity decreases over time
Half-life	$t_{1/2} = \frac{\ln 2}{|k|}$	Time for quantity to halve
Doubling time	$t_{2} = \frac{\ln 2}{k}$	Time for quantity to double