📚 Concept Review & Key Formulas
1. Limits & Continuity
Unit 1- A limit describes the value \(f(x)\) approaches as \(x \to c\), even if \(f(c)\) is undefined.
- A function is continuous at \(c\) if: (i) \(f(c)\) exists, (ii) \(\lim_{x\to c}f(x)\) exists, (iii) both are equal.
\[\lim_{x \to 0} \frac{\sin x}{x} = 1, \qquad \lim_{x \to 0} \frac{1-\cos x}{x} = 0\]
✦ Must Memorize
- L'Hôpital's Rule: if \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), then \(\lim \frac{f}{g} = \lim \frac{f'}{g'}\)
- IVT: if \(f\) is continuous on \([a,b]\) and \(k\) is between \(f(a)\) and \(f(b)\), then \(\exists\, c\) with \(f(c)=k\)
🔍 Example
Find \(\lim_{x\to 0}\dfrac{\tan x}{x}\). Write \(\tan x = \dfrac{\sin x}{\cos x}\), so \(\dfrac{\tan x}{x} = \dfrac{\sin x}{x}\cdot\dfrac{1}{\cos x} \to 1 \cdot 1 = 1\).
2. Differentiation
Unit 2–3\[\frac{d}{dx}[x^n]=nx^{n-1},\quad \frac{d}{dx}[\sin x]=\cos x,\quad \frac{d}{dx}[\cos x]=-\sin x\]
\[\frac{d}{dx}[e^x]=e^x,\quad \frac{d}{dx}[\ln x]=\frac{1}{x},\quad \frac{d}{dx}[\tan x]=\sec^2 x\]
✦ Must Memorize
- Chain Rule: \(\frac{d}{dx}[f(g(x))] = f'(g(x))\cdot g'(x)\)
- Product Rule: \((uv)' = u'v + uv'\)
- Quotient Rule: \(\left(\dfrac{u}{v}\right)' = \dfrac{u'v - uv'}{v^2}\)
- Implicit Differentiation: differentiate both sides w.r.t. \(x\), treat \(y\) as a function of \(x\)
🔍 Example
Differentiate \(f(x)=x^2 e^x\). By Product Rule: \(f'(x)=2xe^x + x^2 e^x = e^x(2x+x^2)\).
3. Applications of Derivatives
Unit 4–5- Critical points: \(f'(c)=0\) or \(f'(c)\) undefined → candidates for local extrema.
- First Derivative Test: \(f'\) changes \(+\to-\) → local max; \(-\to+\) → local min.
- Second Derivative Test: \(f''(c)>0\) → local min; \(f''(c)<0\) → local max.
- Concavity: \(f''>0\) → concave up; \(f''<0\) → concave down. Inflection: sign change of \(f''\).
✦ Must Memorize
- MVT: if \(f\) is continuous on \([a,b]\) and differentiable on \((a,b)\), then \(\exists\, c\) with \(f'(c)=\dfrac{f(b)-f(a)}{b-a}\)
- Rolle's Theorem: if \(f(a)=f(b)\), then \(\exists\, c\) with \(f'(c)=0\)
4. Integration
Unit 6\[\int x^n\,dx = \frac{x^{n+1}}{n+1}+C\;(n\ne -1),\quad \int e^x\,dx=e^x+C,\quad \int \frac{1}{x}\,dx=\ln|x|+C\]
\[\int \sin x\,dx=-\cos x+C,\quad \int \cos x\,dx=\sin x+C,\quad \int \sec^2 x\,dx=\tan x+C\]
✦ Must Memorize
- FTC Part 1: \(\dfrac{d}{dx}\displaystyle\int_a^x f(t)\,dt = f(x)\)
- FTC Part 2: \(\displaystyle\int_a^b f(x)\,dx = F(b)-F(a)\) where \(F'=f\)
- U-substitution: let \(u=g(x)\), then \(du=g'(x)\,dx\)
🔍 Example
Compute \(\displaystyle\int_0^1 2x\,dx\). Antiderivative: \(x^2\Big|_0^1 = 1-0 = 1\).
5. Differential Equations & Area/Volume
Unit 7–8- Separable ODE: separate variables, integrate both sides.
- Exponential Growth/Decay: \(\dfrac{dy}{dt}=ky \Rightarrow y=Ce^{kt}\)
- Area between curves: \(\displaystyle\int_a^b [f(x)-g(x)]\,dx\) (top minus bottom)
- Disk method: \(\displaystyle\pi\int_a^b [f(x)]^2\,dx\); Washer: \(\displaystyle\pi\int_a^b ([R(x)]^2-[r(x)]^2)\,dx\)
6. Sequences, Series & Parametric (BC)
BC Unit 9–10\[e^x = \sum_{n=0}^{\infty}\frac{x^n}{n!},\quad \sin x = \sum_{n=0}^{\infty}\frac{(-1)^n x^{2n+1}}{(2n+1)!},\quad \frac{1}{1-x}=\sum_{n=0}^{\infty}x^n\;(|x|<1)\]
✦ Must Memorize
- Ratio Test: \(\lim_{n\to\infty}\left|\dfrac{a_{n+1}}{a_n}\right|=L\): converges if \(L<1\), diverges if \(L>1\)
- Alternating Series Est.: \(|S-S_n| \le |a_{n+1}|\)
- Arc length (parametric): \(\displaystyle\int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\,dt\)
- Slope of parametric: \(\dfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt}\)
📝 Exam Questions
Question 1
Limits
Easy
What is \(\displaystyle\lim_{x \to 0} \frac{\sin(3x)}{x}\)?
✦ Full Solution
Write \(\dfrac{\sin(3x)}{x} = 3\cdot\dfrac{\sin(3x)}{3x}\). As \(x\to 0\), let \(u=3x\to 0\), so \(\dfrac{\sin(3x)}{3x}\to 1\). Therefore the limit is \(3\cdot 1 = \mathbf{3}\). The correct answer is C.
Question 2
Limits / L'Hôpital
Medium
What is \(\displaystyle\lim_{x \to 0} \frac{e^x - 1 - x}{x^2}\)?
✦ Full Solution
Form \(\frac{0}{0}\) → apply L'Hôpital: \(\dfrac{e^x-1}{2x}\). Still \(\frac{0}{0}\) → apply again: \(\dfrac{e^x}{2}\). As \(x\to 0\): \(\dfrac{e^0}{2}=\dfrac{1}{2}\). The correct answer is B.
Question 3
Continuity
Medium
Let \(f(x) = \begin{cases} \dfrac{x^2 - 4}{x - 2} & x \ne 2 \\ k & x = 2 \end{cases}\). For what value of \(k\) is \(f\) continuous at \(x = 2\)?
✦ Full Solution
Factor: \(\dfrac{x^2-4}{x-2}=\dfrac{(x-2)(x+2)}{x-2}=x+2\) for \(x\ne 2\). So \(\lim_{x\to 2}f(x)=2+2=4\). For continuity, \(k=4\). The correct answer is B.
Question 4
Derivatives — Chain Rule
Medium
If \(f(x) = \sin(x^3)\), then \(f'(x) =\)
✦ Full Solution
Chain Rule: outer function \(\sin(u)\), inner \(u=x^3\). \(f'(x)=\cos(x^3)\cdot 3x^2 = 3x^2\cos(x^3)\). The correct answer is B.
Question 5
Implicit Differentiation
Hard
If \(x^2 + y^2 = 25\), what is \(\dfrac{dy}{dx}\)?
✦ Full Solution
Differentiate both sides w.r.t. \(x\): \(2x + 2y\dfrac{dy}{dx}=0\). Solve: \(\dfrac{dy}{dx}=-\dfrac{2x}{2y}=-\dfrac{x}{y}\). The correct answer is B.
Question 6
Mean Value Theorem
Medium
Let \(f(x) = x^2\) on \([1, 3]\). What value of \(c\) satisfies the conclusion of the Mean Value Theorem?
✦ Full Solution
MVT: \(f'(c)=\dfrac{f(3)-f(1)}{3-1}=\dfrac{9-1}{2}=4\). Since \(f'(x)=2x\), set \(2c=4 \Rightarrow c=2\). The correct answer is B.
Question 7
Related Rates
Hard
A spherical balloon is being inflated so that its volume increases at a rate of \(10\) cm³/s. How fast is the radius increasing when the radius is \(5\) cm? (Volume of sphere: \(V = \frac{4}{3}\pi r^3\))
✦ Full Solution
Differentiate \(V=\frac{4}{3}\pi r^3\) w.r.t. \(t\): \(\dfrac{dV}{dt}=4\pi r^2\dfrac{dr}{dt}\). Plug in \(\dfrac{dV}{dt}=10\), \(r=5\): \(10=4\pi(25)\dfrac{dr}{dt}\), so \(\dfrac{dr}{dt}=\dfrac{10}{100\pi}=\dfrac{1}{10\pi}\) cm/s. The correct answer is A.
Question 8
Definite Integral / FTC
Easy
Evaluate \(\displaystyle\int_0^{\pi} \sin x \, dx\).
✦ Full Solution
\(\int_0^{\pi}\sin x\,dx = [-\cos x]_0^{\pi} = (-\cos\pi)-(-\cos 0)=(-(-1))-(-1)=1+1=2\). The correct answer is C.
Question 9
U-Substitution
Medium
Evaluate \(\displaystyle\int x\cos(x^2)\,dx\).
✦ Full Solution
Let \(u=x^2\), so \(du=2x\,dx\), meaning \(x\,dx=\dfrac{du}{2}\). Integral becomes \(\int\cos u\cdot\dfrac{du}{2}=\dfrac{1}{2}\sin u + C = \dfrac{1}{2}\sin(x^2)+C\). The correct answer is B.
Question 10
FTC Part 1
Medium
Let \(F(x)=\displaystyle\int_1^x \sqrt{t^2+1}\,dt\). What is \(F'(x)\)?
✦ Full Solution
By FTC Part 1: \(\dfrac{d}{dx}\displaystyle\int_1^x f(t)\,dt = f(x)\). Here \(f(t)=\sqrt{t^2+1}\), so \(F'(x)=\sqrt{x^2+1}\). The correct answer is A.
Question 11
Area Between Curves
Medium
What is the area of the region enclosed by \(y = x^2\) and \(y = x\)?
✦ Full Solution
Intersections: \(x^2=x \Rightarrow x=0,1\). On \([0,1]\), \(x \ge x^2\). Area \(=\displaystyle\int_0^1(x-x^2)\,dx=\left[\dfrac{x^2}{2}-\dfrac{x^3}{3}\right]_0^1=\dfrac{1}{2}-\dfrac{1}{3}=\dfrac{1}{6}\). The correct answer is A.
Question 12
Volume of Revolution (Disk)
Hard
The region bounded by \(y = \sqrt{x}\), \(x=4\), and \(y=0\) is revolved about the \(x\)-axis. What is the volume?
✦ Full Solution
Disk method: \(V=\pi\displaystyle\int_0^4(\sqrt{x})^2\,dx=\pi\int_0^4 x\,dx=\pi\left[\dfrac{x^2}{2}\right]_0^4=\pi\cdot\dfrac{16}{2}=8\pi\). The correct answer is B.
Question 13
Differential Equations
Medium
If \(\dfrac{dy}{dx} = 2y\) and \(y(0) = 3\), then \(y =\)
✦ Full Solution
Separable: \(\dfrac{dy}{y}=2\,dx\). Integrate: \(\ln|y|=2x+C_1\), so \(y=Ce^{2x}\). Apply \(y(0)=3\): \(3=Ce^0=C\). Thus \(y=3e^{2x}\). The correct answer is A.
Question 14
Optimization
Hard
A rectangle has one side on the \(x\)-axis and its two upper corners on the parabola \(y = 9 - x^2\). What is the maximum area of such a rectangle?
✦ Full Solution
Let \(x > 0\) be the half-width of the rectangle. Width \(= 2x\), height \(= 9-x^2\). Area: \(A(x)=2x(9-x^2)=18x-2x^3\). Set \(A'(x)=18-6x^2=0 \Rightarrow x^2=3 \Rightarrow x=\sqrt{3}\). Maximum area: \(A(\sqrt{3})=2\sqrt{3}\cdot(9-3)=2\sqrt{3}\cdot 6=12\sqrt{3}\approx 20.8\). Confirm maximum: \(A''(\sqrt{3})=-12\sqrt{3}<0\) ✓. The correct answer is A.
Question 15
Integration by Parts (BC)
Hard
Evaluate \(\displaystyle\int x e^x\,dx\).
✦ Full Solution
Integration by parts: let \(u=x\), \(dv=e^x\,dx\). Then \(du=dx\), \(v=e^x\). Formula: \(\int u\,dv = uv - \int v\,du = xe^x - \int e^x\,dx = xe^x - e^x + C\). Verify by differentiating: \(\frac{d}{dx}[xe^x-e^x]=e^x+xe^x-e^x=xe^x\) ✓. The correct answer is B.
Question 16
Infinite Series — Convergence (BC)
Hard
Which of the following series converges?
✦ Full Solution
A: Harmonic series — diverges (p-series with \(p=1\)). B: p-series with \(p=2>1\) — converges ✓. C: \(\lim_{n\to\infty}\dfrac{n}{n+1}=1\ne 0\) — diverges by Divergence Test. D: terms \(\to\infty\) — diverges. The correct answer is B.
Question 17
Taylor / Maclaurin Series (BC)
Hard
The Maclaurin series for \(e^x\) begins \(1+x+\dfrac{x^2}{2!}+\cdots\). What is the coefficient of \(x^3\) in the Maclaurin series for \(e^{2x}\)?
✦ Full Solution
Replace \(x\) with \(2x\) in \(e^x=\sum_{n=0}^{\infty}\dfrac{x^n}{n!}\): \(e^{2x}=\sum_{n=0}^{\infty}\dfrac{(2x)^n}{n!}=1+2x+\dfrac{(2x)^2}{2!}+\dfrac{(2x)^3}{3!}+\cdots\). The \(x^3\) term is \(\dfrac{2^3\,x^3}{3!}=\dfrac{8x^3}{6}=\dfrac{4}{3}x^3\). Coefficient of \(x^3\) is \(\dfrac{4}{3}\). The correct answer is B.
Question 18
Parametric Curves (BC)
Hard
A curve is defined parametrically by \(x = t^2\) and \(y = t^3\). What is \(\dfrac{dy}{dx}\) at \(t = 2\)?
✦ Full Solution
\(\dfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt}=\dfrac{3t^2}{2t}=\dfrac{3t}{2}\). At \(t=2\): \(\dfrac{3(2)}{2}=3\). The correct answer is A.
Question 19
Polar Area (BC)
Hard
What is the area enclosed by the polar curve \(r = 2\cos\theta\) for \(0 \le \theta \le \pi\)?
✦ Full Solution
Polar area: \(A=\dfrac{1}{2}\displaystyle\int_0^{\pi}r^2\,d\theta=\dfrac{1}{2}\int_0^{\pi}4\cos^2\theta\,d\theta=2\int_0^{\pi}\dfrac{1+\cos 2\theta}{2}\,d\theta=\int_0^{\pi}(1+\cos 2\theta)\,d\theta=\left[\theta+\dfrac{\sin 2\theta}{2}\right]_0^{\pi}=\pi+0=\pi\). The circle \(r=2\cos\theta\) has radius 1, area \(=\pi(1)^2=\pi\) ✓. The correct answer is A.
Question 20
Radius of Convergence (BC)
Hard
What is the radius of convergence of the power series \(\displaystyle\sum_{n=0}^{\infty} \frac{x^n}{n!}\)?
✦ Full Solution
Ratio Test: \(\left|\dfrac{a_{n+1}}{a_n}\right|=\left|\dfrac{x^{n+1}/(n+1)!}{x^n/n!}\right|=\dfrac{|x|}{n+1}\to 0\) as \(n\to\infty\), for all \(x\). Since the limit is \(0 < 1\) for every \(x\), the series converges for all real \(x\). The radius of convergence is \(\infty\). (This is the series for \(e^x\).) The correct answer is D.
FINAL RESULT
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