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- Factoring with leading coefficients \(\neq 1\) and rational expressions
- Negative and fractional exponents rewritten as radicals
- Simplifying compound fractions — backbone of the difference quotient
- Completing the square — curve sketching and integration preview
- Conjugate rationalization for limit problems
Simplify the compound fraction. Show all steps and state values of \(x\) where undefined. \[ \frac{\dfrac{1}{x+h} - \dfrac{1}{x}}{h} \]
Rationalize the denominator of \(f(x)=\dfrac{3}{\sqrt{x+5}-\sqrt{x}}\). Then evaluate \(\displaystyle\lim_{x\to 0}f(x)\).
Rewrite \(g(x)=x^{-2/3}+4x^{1/2}-7x^{-1}\) using only radicals and no negative exponents. Identify the domain of \(g\).
Simplify completely with positive exponents: (a) \(\dfrac{x^2-9}{x^2+x-6}\) (b) \(\dfrac{2x^{-1}+3x^{-2}}{x^{-1}-1}\) (c) \(\left(\dfrac{8x^3}{27y^6}\right)^{-2/3}\)
Factor completely: (a) \(6x^2+7x-3\) (b) \(x^3-8\) (c) \(4x^4-13x^2+9\)
Complete the square for \(f(x)=2x^2-12x+7\). Write in vertex form \(a(x-h)^2+k\) and identify the vertex. Then sketch the parabola.
Rationalize and simplify: \(\dfrac{x-4}{\sqrt{x}-2}\). For what value of \(x\) is the original undefined? What does this suggest about a limit as \(x\to 4\)?
Simplify the compound fraction: \[\dfrac{\dfrac{1}{(x+h)^2}-\dfrac{1}{x^2}}{h}\] Show every step. This is the difference quotient for \(f(x)=\dfrac{1}{x^2}\).
Rewrite each with only positive, fractional exponents (no radicals): (a) \(\sqrt[3]{x^2}\) (b) \(\dfrac{1}{\sqrt[4]{x^3}}\) (c) \(x\sqrt{x}\) (d) \(\dfrac{\sqrt{x}}{\sqrt[3]{x}}\)
Perform the polynomial long division: \((2x^3-3x^2+x-5)\div(x-2)\). State the quotient and remainder. Verify by multiplying back.
Simplify completely. Identify all excluded values of \(x\). \[ \frac{x^2+5x+6}{x^2-4} \cdot \frac{x^2-5x+6}{x^2-9} \]
Rationalize the numerator (not denominator): \(\dfrac{\sqrt{x+h}-\sqrt{x}}{h}\). Take the limit as \(h\to 0\). What derivative did you just compute?
Write 3–4 sentences explaining: (a) Why compound fractions appear in the difference quotient, (b) why completing the square will reappear in integration, and (c) what "indeterminate form \(0/0\)" means and how factoring resolves it.
- Domain restrictions — discontinuities and undefined derivatives
- Piecewise functions: reading, writing, evaluating — AP exam staple
- Transformations at speed: shifting and stretching without plotting
- Absolute value functions and their corners (non-differentiability preview)
- Inverse functions algebraically and graphically
Let \(f(x)=\begin{cases}x^2-1&x<2\\kx+1&x\geq 2\end{cases}\). Find \(k\) so \(f\) is continuous at \(x=2\). Justify your answer.
Let \(g(x)=|2x-6|\). Rewrite as a piecewise function without absolute value notation. Sketch and identify any point where \(g\) may fail to be differentiable.
For \(h(x)=\sqrt{9-x^2}\), find the domain and range. Find \(h^{-1}(x)\) and its domain. Explain geometrically what the graph of \(h^{-1}\) looks like relative to \(h\).
State the domain in interval notation and identify all undefined values.
(a) \(f(x)=\dfrac{\sqrt{x-3}}{x^2-4}\) (b) \(g(x)=\ln(2x+6)\) (c) \(h(x)=\dfrac{1}{\sqrt{9-x^2}}\)
Write the piecewise function whose graph has: slope \(2\) for \(x<0\), horizontal at \(y=3\) for \(0\leq x<4\), slope \(-1\) starting at \((4,3)\) for \(x\geq 4\). Evaluate \(f(-2),f(0),f(3),f(4),f(7)\).
Find the inverse of \(f(x)=\dfrac{2x+3}{x-1}\). State the domain of \(f^{-1}\). Verify algebraically that \(f(f^{-1}(x))=x\).
For \(f(x)=|3x-9|\), write as a piecewise function, find \(f(-1),f(3),f(5)\), and determine at what \(x\)-value \(f\) is not differentiable. Explain why.
Determine whether each function is even, odd, or neither. Show algebraic justification.
(a) \(f(x)=x^4-3x^2\) (b) \(g(x)=x^3+2x\) (c) \(h(x)=x^2+x\)
For \(f(x)=\sqrt{x}\) and \(g(x)=x^2-4\), find and simplify: (a) \(f(g(x))\) and its domain, (b) \(g(f(x))\) and its domain. Are these the same function?
Starting from \(y=x^2\), describe in words the transformations needed to obtain \(y=-2(x+3)^2+5\). Then write the domain and range of the transformed function.
Sketch the graph of \(f(x)=\begin{cases}-x^2+4&x<1\\2x-1&x\geq 1\end{cases}\). Find all \(x\)-intercepts, \(y\)-intercept, and determine whether \(f\) is continuous at \(x=1\). Justify.
Find \(f^{-1}(x)\) for \(f(x)=e^{2x-1}\). State the domain and range of both \(f\) and \(f^{-1}\). Verify by showing \(f(f^{-1}(x))=x\).
On the AP exam, piecewise functions appear in limit and derivative questions. For the function in Problem 8: (a) find \(\displaystyle\lim_{x\to 1^-}f(x)\) and \(\displaystyle\lim_{x\to 1^+}f(x)\), (b) does \(\displaystyle\lim_{x\to 1}f(x)\) exist? (c) Is \(f\) differentiable at \(x=1\)? Explain each answer.
- Properties of exponents and logarithms — full fluency required
- The natural base \(e\) and \(\ln x\): why calculus is built around them
- Solving exponential and logarithmic equations precisely
- Exponential growth/decay models \(A=Pe^{rt}\)
- Graphs of \(e^x\) and \(\ln x\): asymptotes, domain, range, inverse relationship
A population is modeled by \(P(t)=200e^{0.05t}\). Find the time at which the population reaches 500. Express your answer in exact form using \(\ln\).
Solve for \(x\): \(\ln(x^2-3)=\ln(2x)\). Check for extraneous solutions and justify which are valid.
Let \(f(x)=e^{2x}-3\). Find \(f^{-1}(x)\), state its domain, and verify \(f(f^{-1}(x))=x\).
Expand using logarithm properties: \(\ln\!\left(\dfrac{x^3\sqrt{x+1}}{e^{2x}}\right)\). Then condense to a single log: \(2\ln x-\tfrac{1}{2}\ln(x+1)+3\).
Solve exactly. (a) \(3e^{2x}=75\) (b) \(\log_3(x+4)+\log_3(x-2)=3\) (c) \(e^{2x}-5e^x+6=0\) using the substitution \(u=e^x\).
Sketch \(f(x)=2e^{x-1}-3\). Label the \(y\)-intercept, horizontal asymptote, and the point where \(f(x)=0\). State domain and range.
A radioactive substance has half-life 30 years. Write a model \(A(t)=A_0 e^{kt}\). Find \(k\). How long until only 10% of the original remains? Show all steps using \(\ln\).
Prove that \(\log_b x = \dfrac{\ln x}{\ln b}\) using properties of natural logarithms. Then use this to evaluate \(\log_5 100\) exactly and as a decimal.
Solve: (a) \(\ln(e^{3x})=12\) (b) \(e^{\ln(x^2)}=9\) (c) \(\ln(\ln x)=1\). State the domain restriction for each.
Find the inverse of \(g(x)=\ln(x-2)+3\). State its domain and range. Sketch both \(g\) and \(g^{-1}\) on the same axes, labeling the line of symmetry.
Compare \(f(x)=2^x\) and \(g(x)=e^x\) graphically. Find their intersection point(s). Which grows faster for large \(x\)? Justify using a limit.
\$5000 is invested at 4% annual interest. Find the value after 10 years under: (a) annual compounding \(A=P(1+r)^t\), (b) continuous compounding \(A=Pe^{rt}\). How much more does continuous compounding earn?
Differentiation preview: Using the fact that \(\dfrac{d}{dx}[e^x]=e^x\), explain in writing why the derivative of \(f(x)=e^{2x}\) is \(2e^{2x}\) (hint: think about the Chain Rule you will learn in Week 5). Then verify numerically by computing \(\dfrac{e^{2(1+h)}-e^2}{h}\) for \(h=0.001\).
- Unit circle fluency: exact values in radians, no calculator
- Pythagorean, reciprocal, and quotient identities — AP non-negotiables
- Graphs of \(\sin x\), \(\cos x\), \(\tan x\): period, amplitude, phase shift
- Key trig limit preview: \(\displaystyle\lim_{x\to 0}\frac{\sin x}{x}=1\)
- Inverse trig functions \(\arcsin,\arccos,\arctan\) — domains and ranges
Without a calculator, evaluate: \[\sin\!\left(\tfrac{5\pi}{6}\right),\quad\cos\!\left(-\tfrac{2\pi}{3}\right),\quad\tan\!\left(\tfrac{3\pi}{4}\right)\]
If \(\sin\theta=-\dfrac{5}{13}\) and \(\theta\) is in Quadrant III, find \(\cos\theta\), \(\tan\theta\), and \(\sin(2\theta)\). Show all work.
Prove: \(\dfrac{\sin^2 x}{1-\cos x}=1+\cos x\). State any values of \(x\) for which the identity is undefined.
Without a calculator, complete the exact values table for \(\sin\theta\), \(\cos\theta\), \(\tan\theta\) at: \(\theta=0,\,\dfrac{\pi}{6},\,\dfrac{\pi}{4},\,\dfrac{\pi}{3},\,\dfrac{\pi}{2},\,\pi,\,\dfrac{3\pi}{2},\,2\pi\).
Solve on \([0,2\pi)\): (a) \(2\cos^2 x-\cos x-1=0\) (b) \(\sin(2x)=\cos x\). Use identities where needed. List all solutions.
Sketch one full period of \(f(x)=3\sin\!\left(2x-\dfrac{\pi}{3}\right)+1\). Label amplitude, period, phase shift, vertical shift, and all key points.
Prove each identity from scratch — do not use a calculator.
(a) \(\sec^2 x - 1 = \tan^2 x\) (b) \(\dfrac{\cos x}{1-\sin x}=\sec x+\tan x\)
Find all solutions in \([0,2\pi)\): \(2\sin^2 x + 3\sin x + 1 = 0\). Show factoring steps and verify each solution on the unit circle.
Evaluate without a calculator: (a) \(\arcsin\!\left(\tfrac{\sqrt{3}}{2}\right)\) (b) \(\arccos(-1)\) (c) \(\arctan(-\sqrt{3})\) (d) \(\cos(\arcsin\tfrac{3}{5})\)
Use the double angle formulas to find \(\sin(2\theta)\), \(\cos(2\theta)\), and \(\tan(2\theta)\) given that \(\cos\theta=-\dfrac{4}{5}\) and \(\theta\) is in Quadrant II.
Sketch \(f(x)=\cos(x)\) and \(g(x)=\sec(x)=\dfrac{1}{\cos x}\) on the same axes over \([-2\pi,2\pi]\). Label all asymptotes of \(g\) and explain why they occur.
Using a calculator or the squeeze theorem argument, fill in the table and estimate the limit: \[\lim_{x\to 0}\frac{\tan x}{x}\] Use \(x=\pm 0.1,\pm 0.01,\pm 0.001\). What do you get?
The derivatives of trig functions come from \(\lim_{x\to 0}\frac{\sin x}{x}=1\) and \(\lim_{x\to 0}\frac{1-\cos x}{x}=0\). Numerically verify both limits using \(x=0.01\) and \(x=0.001\). Then explain in 2–3 sentences why these two limits are the foundation of all trig differentiation.
- Informal definition: approaching \(c\), not arriving at \(c\)
- One-sided limits: \(\displaystyle\lim_{x\to c^-}\) and \(\displaystyle\lim_{x\to c^+}\)
- When a two-sided limit exists vs. DNE
- Estimating limits from graphs and tables of values
- AP exam notation: writing limit statements correctly
Use the graph of \(f\) to find each value or state DNE. Write a one-sentence justification for each. \[\lim_{x\to 2^-}f(x),\quad\lim_{x\to 2^+}f(x),\quad\lim_{x\to 2}f(x),\quad f(2)\]
Values of \(g(x)\) near \(x=3\): \(x\): 2.9, 2.99, 2.999 → \(g(x)\): 5.71, 5.97, 5.997; \(x\): 3.1, 3.01, 3.001 → \(g(x)\): 6.31, 6.03, 6.003. Estimate \(\displaystyle\lim_{x\to 3}g(x)\) or explain why it does not exist.
Explain the difference between \(\displaystyle\lim_{x\to c}f(x)=L\) and \(f(c)=L\). Give an example where the limit exists at \(x=c\) but \(f(c)\) is undefined.
For \(f(x)=\dfrac{x^2-4}{x-2}\), build a table for \(x=1.9,1.99,1.999,2.001,2.01,2.1\). What does \(\displaystyle\lim_{x\to 2}f(x)\) appear to be? Is \(f(2)\) defined? What type of discontinuity is this?
Sketch a function \(h\) satisfying ALL of: \(\displaystyle\lim_{x\to 1^-}h(x)=3\), \(\displaystyle\lim_{x\to 1^+}h(x)=3\), \(h(1)=5\), \(h\) defined for all real \(x\). Label all key features.
Determine whether \(\displaystyle\lim_{x\to 0}\) exists for each. Justify using one-sided limits.
(a) \(f(x)=\dfrac{|x|}{x}\) (b) \(g(x)=\begin{cases}x+1&x\leq 0\\x^2+1&x>0\end{cases}\)
Using a table of values, estimate each limit to three decimal places. Then verify algebraically if possible.
(a) \(\displaystyle\lim_{x\to 1}\frac{x^3-1}{x-1}\) (b) \(\displaystyle\lim_{x\to 0}\frac{e^x-1}{x}\)
Sketch a function \(f\) defined on \([-3,3]\) that satisfies all of the following simultaneously: (a) \(\displaystyle\lim_{x\to -1}f(x)\) DNE, (b) \(f(-1)=2\) exists, (c) \(\displaystyle\lim_{x\to 2}f(x)=4\) but \(f(2)\) is undefined, (d) \(f\) is continuous at \(x=0\).
For the function \(f(x)=\sin\!\left(\dfrac{1}{x}\right)\), use a table with \(x=1,0.1,0.01,0.001\) and also \(x=\dfrac{2}{\pi},\dfrac{2}{3\pi},\dfrac{2}{5\pi}\). Does \(\displaystyle\lim_{x\to 0}f(x)\) exist? Explain carefully.
Use the graph description: \(f(x)=\begin{cases}2x+1&x<0\\3&x=0\\x^2-1&0
The limit \(\displaystyle\lim_{x\to 2}\frac{x^2-4}{x-2}=4\) using the formal \(\varepsilon\)-\(\delta\) idea (don't prove it formally): explain in plain language what it means for \(f(x)\) to get "arbitrarily close to 4" as \(x\) approaches 2. How small must \(|x-2|\) be to guarantee \(|f(x)-4|<0.1\)?
True or False — justify each answer with a counterexample or explanation.
(a) If \(f(c)\) is defined, then \(\displaystyle\lim_{x\to c}f(x)\) exists.
(b) If \(\displaystyle\lim_{x\to c}f(x)\) exists, then \(f(c)\) is defined.
(c) If \(\displaystyle\lim_{x\to c^-}f(x)=\displaystyle\lim_{x\to c^+}f(x)\), then \(f\) is continuous at \(x=c\).
Write a paragraph explaining the limit concept to a student who has never seen calculus. Use a real-world analogy (speed, temperature, etc.). Then explain why knowing \(\displaystyle\lim_{x\to 2}\frac{x^2-4}{x-2}=4\) is different from just plugging in \(x=2\).
- Strategy 1: Direct substitution — when it works and when it fails
- Strategy 2: Factor and cancel for \(\frac{0}{0}\) indeterminate form
- Strategy 3: Conjugate rationalization for radical expressions
- Strategy 4: Trig limits — \(\displaystyle\lim_{x\to 0}\frac{\sin x}{x}=1\) and \(\displaystyle\lim_{x\to 0}\frac{1-\cos x}{x}=0\)
- The Squeeze Theorem — concept and AP application
Find each limit analytically. Show all steps and identify the strategy used. \[\lim_{x\to 3}\frac{x^2-9}{x-3},\qquad\lim_{x\to 0}\frac{\sqrt{x+4}-2}{x},\qquad\lim_{x\to 0}\frac{\sin(3x)}{x}\]
It is known that \(3x-4\leq g(x)\leq x^2-2x\) for all \(x\) near \(x=1\). Use the Squeeze Theorem to find \(\displaystyle\lim_{x\to 1}g(x)\). Justify your use of the theorem completely.
For \(f(x)=\dfrac{x^2+x-6}{x^2-x-2}\), find all values where \(f\) is undefined. Then find \(\displaystyle\lim_{x\to 2}f(x)\) and \(\displaystyle\lim_{x\to -1}f(x)\) if they exist.
Evaluate each limit. Label the strategy used. \[\lim_{x\to -2}\frac{x^3+8}{x+2},\qquad\lim_{x\to 0}\frac{1-\cos x}{x^2},\qquad\lim_{x\to 9}\frac{\sqrt{x}-3}{x-9}\]
Evaluate \(\displaystyle\lim_{x\to 0}x^2\sin\!\left(\dfrac{1}{x}\right)\) using the Squeeze Theorem. What do you know about \(\sin(1/x)\)? Justify each step carefully.
Find each limit analytically. Show all algebraic steps.
(a) \(\displaystyle\lim_{x\to 4}\frac{x-4}{\sqrt{x}-2}\) (b) \(\displaystyle\lim_{x\to 0}\frac{\sin(5x)}{3x}\) (c) \(\displaystyle\lim_{x\to 1}\frac{x^3-1}{x^2-1}\)
Let \(f(x)=\dfrac{2x^2-5x-3}{x-3}\).
(a) What value of \(x\) makes \(f\) undefined?
(b) Find \(\displaystyle\lim_{x\to 3}f(x)\).
(c) Define \(g(x)\) to agree with \(f(x)\) everywhere except at \(x=3\), where \(g(3)=\displaystyle\lim_{x\to 3}f(x)\). Is \(g\) continuous at \(x=3\)? Justify.
It is given that \(-x^2\leq f(x)\leq x^2\) for all \(x\). Use the Squeeze Theorem to find \(\displaystyle\lim_{x\to 0}f(x)\). Also find \(\displaystyle\lim_{x\to 0}x^2\cos\!\left(\dfrac{1}{x^2}\right)\) by the same method.
Without using L'Hôpital's Rule, evaluate:
(a) \(\displaystyle\lim_{x\to 0}\frac{\tan(2x)}{x}\) (b) \(\displaystyle\lim_{x\to 0}\frac{\sin^2(3x)}{x^2}\) (c) \(\displaystyle\lim_{x\to 0}\frac{x}{\sin(4x)}\)
Determine all values of \(a\) and \(b\) (if any exist) such that \(\displaystyle\lim_{x\to 2}\frac{x^2+ax+b}{x-2}=6\). Hint: for the limit to exist and be finite, what must be true about the numerator at \(x=2\)?
Evaluate: (a) \(\displaystyle\lim_{h\to 0}\frac{(3+h)^2-9}{h}\) (b) \(\displaystyle\lim_{h\to 0}\frac{\sqrt{4+h}-2}{h}\). Recognize what derivative each limit represents and name the function.
Classify each as \(0/0\), \(k/0\), or neither, then evaluate the limit or state it DNE.
(a) \(\displaystyle\lim_{x\to 2}\frac{x^2-4}{x^2-4x+4}\) (b) \(\displaystyle\lim_{x\to 3}\frac{x+1}{x-3}\) (c) \(\displaystyle\lim_{x\to 0}\frac{x^2-5}{x+1}\)
Let \(f(x)=\dfrac{x^2-1}{|x-1|}\).
(a) Find \(\displaystyle\lim_{x\to 1^-}f(x)\) and \(\displaystyle\lim_{x\to 1^+}f(x)\) by rewriting \(|x-1|\) as a piecewise function.
(b) Does \(\displaystyle\lim_{x\to 1}f(x)\) exist?
(c) Is \(f\) continuous at \(x=1\)?
(d) Sketch the graph of \(f\) near \(x=1\).
- Three conditions for continuity: \(f(c)\) defined; limit exists; limit equals \(f(c)\)
- Types of discontinuity: removable, jump, infinite
- Continuity on open and closed intervals
- Intermediate Value Theorem — statement, conditions, AP justification language
- Using IVT to prove existence of zeros and solutions
Let \(f(x)=\begin{cases}\dfrac{x^2-4}{x-2}&x\neq 2\\k&x=2\end{cases}\). Find \(k\) making \(f\) continuous at \(x=2\). Justify using all three conditions for continuity.
Let \(g(x)=x^3-4x+1\). Using the IVT, show \(g\) has at least one zero on \([0,2]\). State all conditions of the IVT explicitly.
Classify each discontinuity as removable, jump, or infinite. Justify.
(i) \(h(x)=\dfrac{x^2-1}{x-1}\) at \(x=1\) (ii) \(p(x)=\dfrac{1}{x-3}\) at \(x=3\) (iii) \(q(x)=\begin{cases}2x&x<1\\x^2&x\geq 1\end{cases}\) at \(x=1\)
Find all discontinuities, classify each, and determine whether it is removable.
(a) \(f(x)=\dfrac{x^2+x-6}{x^2-4}\) (b) \(g(x)=\dfrac{\sin x}{x}\) at \(x=0\) (c) \(h(x)=\dfrac{x+3}{|x+3|}\) at \(x=-3\)
Write a complete IVT justification (paragraph form, as graded on the AP exam) showing that \(f(x)=e^x-3x\) has at least one zero on \([0,1]\). Your justification must reference continuity, endpoint values, and the conclusion.
Find constants \(a\) and \(b\) such that \(f(x)=\begin{cases}ax^2+b&x\leq 1\\2ax-b&x>1\end{cases}\) is continuous everywhere. Show all work.
Use IVT to show that \(f(x)=x^3+x-1\) has exactly one real zero on \([0,1]\). (Hint: first show existence by IVT, then argue uniqueness by noting \(f\) is strictly increasing.)
Let \(f(x)=\begin{cases}x^2-1&x<0\\1&x=0\\2x-1&x>0\end{cases}\). Determine whether \(f\) is continuous at \(x=0\). If not, classify the discontinuity. Can it be made continuous by redefining \(f(0)\)?
True or False — justify with proof or counterexample.
(a) If \(f\) is continuous on \([a,b]\) and \(f(a)>0\) and \(f(b)>0\), then \(f(x)>0\) for all \(x\in[a,b]\).
(b) If \(f\) is differentiable at \(x=c\), then \(f\) is continuous at \(x=c\).
A company's daily profit (in thousands) is modeled by \(P(t)=t^3-6t^2+9t-2\) for \(t\in[0,5]\) (days). Use IVT to show there is a day when profit equals exactly \(\$2000\). State all IVT conditions.
For each function, find all constants that make it continuous everywhere.
(a) \(f(x)=\begin{cases}3x+k&x<2\\x^2+1&x\geq 2\end{cases}\) (b) \(g(x)=\begin{cases}ax+b&x<1\\3&x=1\\bx^2+a&x>1\end{cases}\)
Explain in your own words the difference between removable, jump, and infinite discontinuities. Give an original example of each — not from the textbook or this homework set. Sketch all three graphs.
The IVT is used on the AP exam to justify the existence of values, not to find them. Write two original IVT justification problems (and their solutions) that could appear on the AP exam. One should involve a polynomial; the other should involve \(e^x\) or \(\ln x\).
- Limits as \(x\to\pm\infty\) — the degree-comparison shortcut
- Horizontal asymptotes defined rigorously through limits
- Vertical asymptotes: \(\displaystyle\lim_{x\to c^\pm}f(x)=\pm\infty\)
- Comparing growth rates: polynomial vs. exponential vs. logarithmic
- End behavior of \(xe^{-x}\), \(\dfrac{\ln x}{x}\), \(x\sin\!\left(\tfrac{1}{x}\right)\) as \(x\to\infty\)
Find each limit. Show all work — no calculator. \[\lim_{x\to\infty}\frac{4x^3-2x+1}{7x^3+5x^2},\qquad\lim_{x\to\infty}\frac{e^x}{x^{100}},\qquad\lim_{x\to\infty}x\sin\!\left(\frac{1}{x}\right)\]
For \(f(x)=\dfrac{3x^2-1}{x^2-4}\), find all horizontal and vertical asymptotes. Justify each using limits. Determine whether \(f\) crosses its horizontal asymptote.
Without a calculator, determine which grows faster as \(x\to\infty\): \(x^{1000}\) or \(e^x\). Justify by evaluating \(\displaystyle\lim_{x\to\infty}\dfrac{x^{1000}}{e^x}\).
Find all horizontal asymptotes (justify with limits) and vertical asymptotes for each function.
(a) \(f(x)=\dfrac{2x^2-3}{x^2+1}\) (b) \(g(x)=\dfrac{x}{\sqrt{x^2+1}}\) (c) \(h(x)=e^{-x^2}\)
Evaluate: (a) \(\displaystyle\lim_{x\to 3^-}\frac{1}{x-3}\) (b) \(\displaystyle\lim_{x\to 3^+}\frac{1}{x-3}\) (c) \(\displaystyle\lim_{x\to 3}\frac{1}{x-3}\). What do the one-sided limits tell you about the two-sided limit?
Note that \(g(x)=\dfrac{x}{\sqrt{x^2+1}}\) from Problem 1(b) has TWO different horizontal asymptotes. Evaluate \(\displaystyle\lim_{x\to\infty}g(x)\) and \(\displaystyle\lim_{x\to-\infty}g(x)\) separately. Explain what this means for the graph. (Hint: \(\sqrt{x^2}=|x|\).)
Evaluate each limit using degree comparison or algebra.
(a) \(\displaystyle\lim_{x\to\infty}\frac{5x^4-3x}{2x^4+7}\) (b) \(\displaystyle\lim_{x\to\infty}\frac{x^3+1}{x^4-1}\) (c) \(\displaystyle\lim_{x\to\infty}\frac{(2x-1)^3}{8x^3+5}\)
Evaluate: (a) \(\displaystyle\lim_{x\to\infty}xe^{-x}\) (use the fact that \(e^x\) grows faster than any power of \(x\)) (b) \(\displaystyle\lim_{x\to\infty}\frac{\ln x}{x}\) (c) \(\displaystyle\lim_{x\to 0^+}x\ln x\)
Sketch a function that has: a horizontal asymptote at \(y=2\) as \(x\to\infty\), a horizontal asymptote at \(y=-1\) as \(x\to-\infty\), and vertical asymptotes at \(x=-3\) and \(x=1\). Write a possible formula for such a function.
For \(f(x)=\dfrac{x^2-4}{x^2-4x+4}\), find all asymptotes. Explain why this function has a vertical asymptote at \(x=2\) even though the numerator also equals zero at \(x=2\). What is different from a removable discontinuity?
Rank the following from slowest to fastest growth as \(x\to\infty\): \(\ln x,\; x^{0.001},\; x^{100},\; e^x,\; 2^x,\; x!\) (for large integer \(x\)). Justify at least three of your rankings using limits.
Evaluate \(\displaystyle\lim_{x\to\infty}\left(\sqrt{x^2+x}-x\right)\). This requires algebra — multiply by the conjugate \(\dfrac{\sqrt{x^2+x}+x}{\sqrt{x^2+x}+x}\). Show every step.
The AP exam often gives a graph and asks about end behavior and asymptotes. Create your own rational function \(f(x)\) that has: (a) a horizontal asymptote at \(y=3\), (b) exactly two vertical asymptotes, (c) at least one \(x\)-intercept. Find all asymptotes and intercepts analytically, then sketch the graph.
- Average rate of change as slope of the secant line
- Instantaneous rate of change: \(f'(x)=\displaystyle\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}\)
- Differentiability vs. continuity — necessary but not sufficient
- When \(f\) is NOT differentiable: corners, cusps, vertical tangents, holes
- Tangent line equation using point-slope form: \(y-f(a)=f'(a)(x-a)\)
Using the limit definition, find \(f'(x)\) for \(f(x)=3x^2-5x+1\). Show all steps — do not use derivative rules.
Find the equation of the tangent line to \(f(x)=\sqrt{x+3}\) at \(x=1\). Use the limit definition to find the slope.
Let \(g(x)=|x-2|\). Is \(g\) differentiable at \(x=2\)? Justify using one-sided limits of the difference quotient. What does this mean geometrically?
Using \(f'(x)=\displaystyle\lim_{h\to 0}\dfrac{f(x+h)-f(x)}{h}\), find \(f'(x)\) for each. Show every algebra step.
(a) \(f(x)=x^3\) (b) \(f(x)=\dfrac{1}{x}\) (c) \(f(x)=\sqrt{x}\)
Using the definition, find \(f'(a)\) at the specific point given.
(a) \(f(x)=x^2-3x\), \(a=2\) (b) \(f(x)=\dfrac{1}{x+1}\), \(a=0\) (c) \(f(x)=\sqrt{2x+1}\), \(a=4\)
The position of a particle is \(s(t)=t^2-4t+3\).
(a) Find the average velocity on \([1,3]\).
(b) Find the instantaneous velocity at \(t=1\) using the limit definition.
(c) At what time is the particle at rest? Moving forward? Moving backward?
Find the equation of the tangent line and the normal line (perpendicular to the tangent) to \(f(x)=x^2+1\) at \(x=2\). Sketch both lines and the curve.
Sketch a function that is continuous everywhere on \([-3,3]\) but not differentiable at exactly two points. Label each non-differentiable point and explain the graph feature that causes it.
For \(f(x)=|x^2-4|\), identify all points where \(f\) is not differentiable. Justify each answer using the definition of the derivative (one-sided limits of the difference quotient).
Using the alternate form of the limit definition \(f'(a)=\displaystyle\lim_{x\to a}\dfrac{f(x)-f(a)}{x-a}\), find \(f'(2)\) for \(f(x)=x^3-x\). Show complete algebraic simplification before taking the limit.
The line \(y=3x-1\) is tangent to \(f(x)=x^2+bx+c\) at \(x=2\). Find \(b\) and \(c\). (Hint: the tangent line must touch the curve AND have the same slope as \(f'\) at that point.)
True or False — justify each:
(a) If \(f\) is differentiable at \(x=c\), then \(f\) is continuous at \(x=c\).
(b) If \(f\) is continuous at \(x=c\), then \(f\) is differentiable at \(x=c\).
(c) If \(f'(c)=0\), then \(f\) has a local max or min at \(x=c\).
The AP exam presents the derivative both graphically and analytically. (a) Given a graph of \(f\), sketch what \(f'\) might look like — label where \(f'>0\), \(f'<0\), and \(f'=0\). (b) Compute \(\displaystyle\lim_{h\to 0}\dfrac{\sin(h)}{h}\) numerically using \(h=0.1,0.01,0.001\). What derivative does this limit represent?
- Power Rule: \(\dfrac{d}{dx}[x^n]=nx^{n-1}\) for all real \(n\)
- Product Rule: \((fg)'=f'g+fg'\)
- Quotient Rule: \(\left(\dfrac{f}{g}\right)'=\dfrac{f'g-fg'}{g^2}\)
- Chain Rule: \(\dfrac{d}{dx}[f(g(x))]=f'(g(x))\cdot g'(x)\) — the most tested rule
- Derivatives of \(e^x,\ln x,\sin x,\cos x,\tan x\) and all six trig functions
Find \(f'(x)\) for each. Identify which rule(s) you used.
(i) \(f(x)=x^3e^{2x}\) (ii) \(g(x)=\dfrac{\ln x}{\sin x}\) (iii) \(h(x)=\sin^3(x^2+1)\)
Let \(f\) and \(g\) be differentiable with \(f(2)=3,f'(2)=5,g(2)=2,g'(2)=4\). Find: \[\frac{d}{dx}[f(x)g(x)]\bigg|_{x=2},\quad\frac{d}{dx}\!\left[\frac{f(x)}{g(x)}\right]\bigg|_{x=2},\quad\frac{d}{dx}[f(g(x))]\bigg|_{x=2}\]
Find the equation of the tangent line to \(h(x)=e^{\sin x}\) at \(x=0\). Then determine whether the graph of \(h\) is concave up or concave down at \(x=0\) by computing \(h''(0)\).
Differentiate. Label each rule used.
(a) \(f(x)=(3x^2-1)^5\) (b) \(g(x)=x^2\ln(x^2+1)\) (c) \(h(x)=\dfrac{e^{2x}}{\cos x}\) (d) \(p(x)=\tan(\sqrt{x})\)
Find \(f'(x)\) for each using the appropriate rule. Do not simplify unless directed.
(a) \(f(x)=\sin(x^3)\cos(x^2)\) (b) \(g(x)=\dfrac{(x^2+1)^3}{e^x}\) (c) \(h(x)=\ln(\sin^2 x+1)\)
Find all values of \(x\) where the tangent line to \(f(x)=x^3-3x^2-9x+5\) is horizontal. At each point determine whether \(f\) has a local max, local min, or neither.
The table gives values of differentiable functions \(f\) and \(g\):
\(x=1\): \(f=2,f'=3,g=3,g'=-1\); \(x=2\): \(f=1,f'=5,g=3,g'=2\); \(x=3\): \(f=4,f'=-2,g=1,g'=7\).
Find: (a) \(\dfrac{d}{dx}[f(g(x))]\big|_{x=1}\) (b) \(\dfrac{d}{dx}[g(f(x))]\big|_{x=2}\) (c) \(\dfrac{d}{dx}[f(x)\cdot g(x)]\big|_{x=3}\)
Find the second derivative \(f''(x)\) for each.
(a) \(f(x)=x^4-3x^2+5\) (b) \(g(x)=\sin(2x)\) (c) \(h(x)=xe^x\)
Differentiate using the chain rule. Show the outer and inner functions before differentiating.
(a) \(y=e^{x^2+3x}\) (b) \(y=\ln(\cos x)\) (c) \(y=\sqrt{\tan x}\) (d) \(y=(\sin x+\cos x)^4\)
A particle moves along a line with position \(s(t)=t^3-6t^2+9t\).
(a) Find velocity \(v(t)=s'(t)\) and acceleration \(a(t)=s''(t)\).
(b) When is the particle at rest? Moving in the positive direction?
(c) When is the particle speeding up? Slowing down?
Find the equation of the tangent line to each curve at the given point.
(a) \(f(x)=x^2e^x\) at \(x=1\) (b) \(g(x)=\dfrac{\sin x}{x}\) at \(x=\pi\)
Find \(f'(x)\) for \(f(x)=\sin(\cos(\tan x))\). Identify every chain rule application and label each outer and inner function at each stage.
Using the table from Problem 4, find: (a) \(\dfrac{d}{dx}\!\left[\dfrac{f(x)}{g(x)}\right]\bigg|_{x=2}\), (b) \(\dfrac{d}{dx}[f(x)^2]\big|_{x=3}\), (c) the equation of the tangent line to \(y=f(g(x))\) at \(x=1\). Show the rule setup for each before substituting.
- Implicit differentiation: differentiating both sides with respect to \(x\)
- Solving for \(\dfrac{dy}{dx}\) algebraically — isolating the derivative
- Tangent and normal lines on implicitly defined curves
- Derivatives of \(\arcsin x\), \(\arccos x\), \(\arctan x\) — derived from implicit differentiation
- Combining implicit differentiation with inverse trig on the same problem
For the curve defined by \(x^2 + 3xy + y^2 = 7\), find \(\dfrac{dy}{dx}\) using implicit differentiation. Then find the equation of the tangent line at the point \((1, 1)\) and the normal line at the same point.
Find \(\dfrac{d}{dx}[\arctan x]\) by starting from \(y = \arctan x\), rewriting as \(\tan y = x\), and differentiating implicitly. Show every step — this is how the formula is derived.
Differentiate each function. Identify any rules beyond the inverse trig derivative formula.
(i) \(f(x) = \arcsin(3x^2)\) (ii) \(g(x) = x^2\arctan(x)\) (iii) \(h(x) = \arccos\!\left(\dfrac{1}{x}\right)\)
Find \(\dfrac{dy}{dx}\) for each using implicit differentiation.
(a) \(x^3 + y^3 = 6xy\) (b) \(\sin(xy) = x\) (c) \(e^{x+y} = x^2y\) (d) \(x^2y + y^3 = 5\)
For the curve \(x^2 - xy + y^2 = 3\), find all points where the tangent line is horizontal (i.e., \(\dfrac{dy}{dx} = 0\)) and all points where the tangent line is vertical (i.e., \(\dfrac{dy}{dx}\) is undefined). Show complete algebraic work.
For the curve \(y^2 = x^3 - x\), find \(\dfrac{dy}{dx}\) implicitly. Then find the equation of the tangent line at the point \((2, \sqrt{6})\). Leave in exact form.
Derive the formula \(\dfrac{d}{dx}[\arcsin x] = \dfrac{1}{\sqrt{1-x^2}}\) from scratch using implicit differentiation. Start with \(y = \arcsin x\), rewrite as \(\sin y = x\), differentiate implicitly, and simplify using a right triangle or Pythagorean identity.
Differentiate each inverse trig function. Show the chain rule setup for each.
(a) \(f(x) = \arctan(e^x)\) (b) \(g(x) = \arcsin(\sqrt{x})\) (c) \(h(x) = \arccos(2x-1)\) (d) \(p(x) = \arctan\!\left(\dfrac{x}{a}\right)\) where \(a\) is a constant
Find \(\dfrac{dy}{dx}\) for each. These combine implicit differentiation with inverse trig.
(a) \(\arctan(xy) = x + y\) (b) \(y = \arcsin(x + y)\)
Find \(\dfrac{d^2y}{dx^2}\) for \(x^2 + y^2 = 25\) by differentiating implicitly twice. Express your final answer in terms of \(x\) and \(y\) only — eliminate \(\dfrac{dy}{dx}\) from the final answer. What does the sign of \(\dfrac{d^2y}{dx^2}\) tell you about the circle?
Differentiate each expression completely. These combine multiple rules.
(a) \(f(x) = x\arctan x - \dfrac{1}{2}\ln(1+x^2)\) (b) \(g(x) = \sqrt{1-x^2}\arcsin x\)
The equation \(x^4 + y^4 = 16\) defines a curve called a "superellipse." Find all points on the curve where the tangent line has slope \(-1\). Show the implicit differentiation setup and all algebraic steps to find the exact coordinates.
On the AP exam, implicit differentiation appears in two contexts: (1) finding \(\dfrac{dy}{dx}\) on a curve, and (2) deriving formulas. For this problem: (a) use implicit differentiation to show that \(\dfrac{d}{dx}[\arctan x]=\dfrac{1}{1+x^2}\), (b) use this formula to evaluate \(\dfrac{d}{dx}\!\left[\arctan\!\left(\dfrac{2x}{1-x^2}\right)\right]\) by chain rule, and (c) verify your answer makes sense by noting that \(\arctan\!\left(\dfrac{2x}{1-x^2}\right) = 2\arctan x\) for \(|x|<1\) and differentiating the right side directly.
- Logarithmic differentiation: taking \(\ln\) of both sides before differentiating
- Differentiating variable-exponent functions like \(x^x\) and \(x^{\sin x}\)
- Second and higher-order derivatives: notation \(f'',f''',f^{(n)}\)
- Position, velocity, acceleration, and jerk from repeated differentiation
- Linear approximation: \(L(x)=f(a)+f'(a)(x-a)\) — tangent line as an approximator
Answer each part completely. Show all rules and steps. Justify every conclusion.
Let \(f(x) = x^x\). Use logarithmic differentiation to find \(f'(x)\). Then find \(f'(1)\) and \(f'(2)\). Explain why the Power Rule alone cannot be used here.
Let \(g(x) = e^{\sin x}\cdot\arctan(x^2)\). Find \(g'(x)\). Identify every differentiation rule used and the order in which you applied them.
The position of a particle moving along a line is given by \(s(t) = t^4 - 8t^2 + 3\) for \(t \geq 0\). Find the velocity \(v(t)\), acceleration \(a(t)\), and jerk \(j(t) = s'''(t)\). Determine the times when the particle is at rest and when it is moving in the positive direction.
Use linear approximation to estimate \(\sqrt{4.02}\). Use \(f(x) = \sqrt{x}\) with \(a = 4\). Write the linearization \(L(x)\), evaluate \(L(4.02)\), and explain why this estimate is close to the true value.
Use logarithmic differentiation to find \(\dfrac{dy}{dx}\) for each. Show the step of taking \(\ln\) of both sides before differentiating.
(a) \(y = x^{\sin x}\) (b) \(y = (\ln x)^x\) (c) \(y = \dfrac{x^3(x+1)^4}{(2x-1)^5}\) (d) \(y = x^{1/x}\)
Explain why logarithmic differentiation is necessary for \(y = x^x\) but NOT necessary for \(y = x^3\) or \(y = 3^x\). What is different about \(x^x\) that makes the Power Rule and the Exponential Rule both fail?
Find the indicated derivative for each.
(a) \(f(x) = x^5 - 3x^3 + 2x\), find \(f^{(4)}(x)\) (b) \(g(x) = \sin(2x)\), find \(g^{(6)}(x)\) (c) \(h(x) = xe^x\), find \(h^{(3)}(x)\)
Find a general formula for \(f^{(n)}(x)\) for each function by computing the first several derivatives and identifying the pattern.
(a) \(f(x) = e^{2x}\) (b) \(f(x) = \sin x\) (c) \(f(x) = \dfrac{1}{x}\)
A particle moves along a line with position \(s(t) = 2t^3 - 9t^2 + 12t - 4\) for \(t \geq 0\).
(a) Find \(v(t)\) and \(a(t)\).
(b) When is the particle at rest? When is it moving in the positive direction?
(c) When is the particle speeding up? (Hint: speeding up means \(v\) and \(a\) have the same sign.)
(d) Find the total distance traveled on \([0, 3]\).
Use linear approximation \(L(x) = f(a) + f'(a)(x-a)\) to estimate each value. State \(f(x)\), \(a\), and show the full setup.
(a) \(\sqrt[3]{8.1}\) (b) \(\ln(1.05)\) (c) \(\sin(0.1)\) (in radians) (d) \(e^{0.03}\)
For \(f(x) = \ln x\), write the linearization \(L(x)\) at \(a = 1\). Use it to estimate \(\ln(1.2)\) and \(\ln(1.8)\). Use a calculator to find the actual values. For which estimate is the linear approximation more accurate, and why?
Find \(\dfrac{dy}{dx}\) for each using the most efficient method — choose between logarithmic differentiation, implicit differentiation, and standard rules.
(a) \(y = \dfrac{(x^2+1)^3\sqrt{x+4}}{e^x(3x-1)^2}\) (b) \(y^x = x^y\) (implicit + log) (c) \(y = (\cos x)^{\tan x}\)
The second derivative test for concavity: for each function, find \(f''(x)\) and all values of \(x\) where \(f''(x) = 0\) or \(f''(x)\) is undefined. These are candidate inflection points — you will fully analyze them in the next phase of your AP prep.
(a) \(f(x) = x^4 - 4x^3\) (b) \(g(x) = xe^{-x}\) (c) \(h(x) = \ln(x^2+1)\)
Write a 1-page self-assessment covering the full six weeks: (a) List the three differentiation rules you feel most confident with and give an example of each from the homework. (b) List the two topics that still feel shaky — be specific with problem numbers. (c) You now know limits, continuity, and derivatives. Write 3–4 sentences explaining how these three ideas are connected to each other. (d) What are you most looking forward to learning next in AP Calculus?
Know your exam: starting May 2027, the AP Calculus AB exam is a hybrid digital exam — 42 multiple-choice questions in Bluebook (Part A: 29 questions, 62 minutes, no calculator; Part B: 13 questions, 38 minutes, graphing calculator required) plus 6 handwritten free-response questions in 90 minutes. Each section is 50% of your score. We will train pacing against these exact numbers during the school year.
Final step: take the End of Boot Camp Assessment → It covers all 12 modules plus foundational material, and Professor Alan will walk you through your full skill map on your results call.
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