Calculus practice by topic

01Prerequisites: lines, exponents, logs, trig12 units

1. 1-1Slope of a line$\displaystyle a=\frac{y_2-y_1}{x_2-x_1}\quad(x_1\neq x_2)$6 examples · 5 patterns→
2. 1-2Function notation f(x) and substitution$\displaystyle f(x)=x^2\ \Rightarrow\ f(-4)=(-4)^2=16$6 examples · 3 patterns→
3. 1-3Equation of a line, y=a(x−p)+q$\displaystyle y=a(x-p)+q$6 examples · 6 patterns→
4. 1-4Expanding (a+b)² and (a+b)³$\displaystyle (a+b)^2=a^2+2ab+b^2$6 examples · 5 patterns→
5. 1-5Factoring out and cancelling$\displaystyle \frac{ah^2+bh}{h}=\frac{h(ah+b)}{h}=ah+b\quad(h\neq 0)$6 examples · 3 patterns→
6. 1-6Solving quadratic equations$\displaystyle ax^2+bx+c=0\ (a\neq 0)\ \Rightarrow\ x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$6 examples · 4 patterns→
7. 1-7Radicals and rationalizing the denominator$\displaystyle (\sqrt{a}+b)(\sqrt{a}-b)=a-b^2$6 examples · 4 patterns→
8. 1-8Translating a graph$\displaystyle y=(x-p)^2+q$6 examples · 6 patterns→
9. 1-9Exponent rules and exponential functions$\displaystyle a^m a^n=a^{m+n}$6 examples · 5 patterns→
10. 1-10Trigonometric functions and radians$\displaystyle 180^\circ=\pi\ \text{rad}$6 examples · 6 patterns→
11. 1-11Trig identities and double angles$\displaystyle \sin^2\theta+\cos^2\theta=1$6 examples · 6 patterns→
12. 1-12Definition of the logarithm$\displaystyle \log_a b=c \iff a^c=b$6 examples · 6 patterns→

02Derivatives: the basics14 units

1. 2-1Average rate of change$\displaystyle \frac{f(b)-f(a)}{b-a}$6 examples · 1 pattern→
2. 2-2Shrinking the interval (h→0)$\displaystyle \frac{(a+h)^2-a^2}{h}=2a+h\ \to\ 2a\quad(h\to 0)$6 examples · 3 patterns→
3. 2-3Difference quotient (and why you cannot divide by 0)$\displaystyle \frac{f(x+h)-f(x)}{h}\quad(h\neq 0)$6 examples · 6 patterns→
4. 2-4The derivative at a point, f'(a)$\displaystyle f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$5 examples · 5 patterns→
5. 2-5The derivative function f'(x)$\displaystyle f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$6 examples · 4 patterns→
6. 2-6xⁿ from the definition$\displaystyle (x^n)'=nx^{n-1}$5 examples · 2 patterns→
7. 2-7Power rule, constant multiples, sums and differences$\displaystyle (x^n)'=nx^{n-1}$5 examples · 2 patterns→
8. 2-8Equation of a tangent line$\displaystyle y=f'(a)(x-a)+f(a)$6 examples · 3 patterns→
9. 2-9Tangent lines from a point not on the curve$\displaystyle y-f(t)=f'(t)(x-t)\quad\Rightarrow\quad q-f(t)=f'(t)(p-t)$6 examples · 6 patterns→
10. 2-10Sign of f' and increasing/decreasing$\displaystyle f'(x)>0 \Rightarrow \text{increasing}$6 examples · 3 patterns→
11. 2-11Local maxima and minima$\displaystyle f'(a)=0,\ f':\ +\to -\ \Rightarrow\ \text{local max}$6 examples · 3 patterns→
12. 2-12Sketching the graph of a cubic$\displaystyle \text{sign of } f':\ +\to-\ \Rightarrow\ \text{local max}$4 examples · 2 patterns→
13. 2-13Maximum and minimum on a closed interval$\displaystyle \text{candidates}:\ f(a),\ f(b),\ f(x_0)$6 examples · 4 patterns→
14. 2-14Number of real solutions of an equation$\displaystyle (\text{number of real solutions of } f(x)=k)$6 examples · 4 patterns→

03Integrals: the basics14 units

1. 3-1Antiderivatives (the reverse of differentiation)$\displaystyle F'(x)=f(x) \iff \int f(x)\,dx=F(x)+C$6 examples · 3 patterns→
2. 3-2The formula for ∫xⁿ dx$\displaystyle \int x^n\,dx=\frac{x^{n+1}}{n+1}+C\quad(n\neq -1)$5 examples · 2 patterns→
3. 3-3Constant multiples, sums and differences (polynomials)$\displaystyle \int kf\,dx=k\int f\,dx$6 examples · 3 patterns→
4. 3-4Approximating area with rectangles$\displaystyle S\approx\sum_{k=0}^{n-1}f(x_k)\,\Delta x$5 examples · 2 patterns→
5. 3-5Area and antiderivatives (the FTC intuition)$\displaystyle S(x)=\int_0^x f(t)\,dt \;\Rightarrow\; S'(x)=f(x)$6 examples · 4 patterns→
6. 3-6Definite integrals: F(b) − F(a)$\displaystyle \int_a^b f(x)\,dx=\Big[F(x)\Big]_a^b=F(b)-F(a)$6 examples · 5 patterns→
7. 3-7Properties of definite integrals$\displaystyle \int_b^a f(x)\,dx=-\int_a^b f(x)\,dx$6 examples · 4 patterns→
8. 3-8Signed area$\displaystyle f(x)\le 0 \text{ on } [a,b] \Rightarrow \int_a^b f(x)\,dx\le 0$6 examples · 1 pattern→
9. 3-9Definite integrals of even and odd functions$\displaystyle \int_{-a}^{a} f(x)\,dx=\begin{cases} 2\int_{0}^{a} f(x)\,dx & (\text{even}) \\ 0 & (\text{odd}) \end{cases}$6 examples · 4 patterns→
10. 3-10Area between two curves$\displaystyle S=\int_a^b \{f(x)-g(x)\}\,dx\quad(f(x)\ge g(x))$6 examples · 3 patterns→
11. 3-11Find the intersections, then the area$\displaystyle f(x)=g(x)\ \Rightarrow\ x=\alpha,\ \beta$6 examples · 4 patterns→
12. 3-12The 1/6 formula$\displaystyle \int_\alpha^\beta (x-\alpha)(x-\beta)\,dx=-\frac{(\beta-\alpha)^3}{6}$6 examples · 3 patterns→
13. 3-13Differentiating an integral with variable upper limit$\displaystyle \frac{d}{dx}\int_a^x f(t)\,dt=f(x)\quad(a\ \text{constant})$6 examples · 5 patterns→
14. 3-14Finding a function defined with a definite integral$\displaystyle f(x)=g(x)+\int_a^b f(t)\,dt\ \Rightarrow\ \text{set } \int_a^b f(t)\,dt=k\ (\text{a constant})$6 examples · 4 patterns→

04Derivatives: rules and functions31 units

1. 4-1Limits of sequences$\displaystyle \lim_{n\to\infty}\frac{1}{n}=0$6 examples · 4 patterns→
2. 4-2Infinite geometric series$\displaystyle \sum_{n=1}^{\infty}ar^{n-1}=\frac{a}{1-r}\quad(|r|<1)$6 examples · 3 patterns→
3. 4-3Graphs of rational functions$\displaystyle y=\frac{k}{x-p}+q\quad(k\neq 0):\quad \text{asymptotes } x=p,\ y=q$6 examples · 4 patterns→
4. 4-4Graphs of irrational functions$\displaystyle y=a\sqrt{x-p}+q\quad(a\neq 0):\quad x\ge p$6 examples · 5 patterns→
5. 4-5What an inverse function means$\displaystyle f(a)=b\iff f^{-1}(b)=a$6 examples · 2 patterns→
6. 4-6Limits of the form 0/0$\displaystyle \lim_{x\to a}\frac{(x-a)g(x)}{x-a}=\lim_{x\to a}g(x)$6 examples · 5 patterns→
7. 4-7Limits as x→∞ (convergence and divergence)$\displaystyle \lim_{x\to\infty}\frac{ax^m+\cdots}{bx^n+\cdots}=\begin{cases}0 & (m<n)\\ \frac{a}{b} & (m=n)\\ \pm\infty & (m>n)\end{cases}$6 examples · 3 patterns→
8. 4-8Asymptotes$\displaystyle y=\frac{k}{x-p}+q\quad(k\neq 0)\ \Rightarrow\ x=p,\ \ y=q$6 examples · 3 patterns→
9. 4-9Continuity and one-sided limits$\displaystyle \lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=f(a)$6 examples · 3 patterns→
10. 4-10Differentiability$\displaystyle \lim_{h\to 0^{+}}\frac{f(a+h)-f(a)}{h}=\lim_{h\to 0^{-}}\frac{f(a+h)-f(a)}{h}$6 examples · 4 patterns→
11. 4-11The limit of sin x/x$\displaystyle \lim_{x\to 0}\frac{\sin x}{x}=1$6 examples · 3 patterns→
12. 4-12Derivatives of sin x and cos x$\displaystyle (\sin x)'=\cos x$6 examples · 6 patterns→
13. 4-13Derivative of tan x$\displaystyle (\tan x)'=\frac{1}{\cos^2 x}$6 examples · 6 patterns→
14. 4-14Derivative of eˣ$\displaystyle (e^x)'=e^x,\qquad (a^x)'=a^x\log a$6 examples · 6 patterns→
15. 4-15Derivative of log x$\displaystyle (\log x)'=\frac{1}{x}$6 examples · 6 patterns→
16. 4-16Product rule$\displaystyle (fg)'=f'g+fg'$6 examples · 6 patterns→
17. 4-17Quotient rule$\displaystyle \left(\frac{f}{g}\right)'=\frac{f'g-fg'}{g^2}$6 examples · 2 patterns→
18. 4-18Meaning of a composite function f(g(x))$\displaystyle (f\circ g)(x)=f(g(x))$6 examples · 3 patterns→
19. 4-19Chain rule$\displaystyle \{f(g(x))\}'=f'(g(x))\,g'(x)$6 examples · 6 patterns→
20. 4-20Derivative of an inverse function$\displaystyle \frac{dy}{dx}=\frac{1}{\,\dfrac{dx}{dy}\,}$6 examples · 4 patterns→
21. 4-21Derivatives of 1/x and √x$\displaystyle (x^r)'=rx^{r-1}$6 examples · 3 patterns→
22. 4-22Logarithmic differentiation$\displaystyle (\log y)'=\frac{y'}{y}$6 examples · 6 patterns→
23. 4-23Second derivative$\displaystyle f''(x)=\{f'(x)\}'$6 examples · 3 patterns→
24. 4-24Concavity and inflection points$\displaystyle f''(x)>0 \Rightarrow \text{concave up}$6 examples · 4 patterns→
25. 4-25Increase, decrease and graph shape (exp and log functions)$\displaystyle (x^n e^{-x})'=x^{n-1}(n-x)e^{-x}$6 examples · 5 patterns→
26. 4-26Differentiating parametric equations$\displaystyle \frac{dy}{dx}=\frac{dy/dt}{dx/dt}\quad\left(\frac{dx}{dt}\neq 0\right)$5 examples · 4 patterns→
27. 4-27Implicit differentiation$\displaystyle \frac{d}{dx}f(y)=f'(y)\,\frac{dy}{dx}$6 examples · 3 patterns→
28. 4-28Normal lines$\displaystyle y-f(a)=-\frac{1}{f'(a)}(x-a)\quad(f'(a)\ne 0)$6 examples · 4 patterns→
29. 4-29Velocity and acceleration$\displaystyle v=\frac{dx}{dt}$6 examples · 4 patterns→
30. 4-30Linear approximation$\displaystyle f(a+h)\approx f(a)+f'(a)h$6 examples · 4 patterns→
31. 4-31Mean value theorem$\displaystyle \frac{f(b)-f(a)}{b-a}=f'(c)\quad(a<c<b)$6 examples · 4 patterns→

05Integrals: functions and by parts14 units

1. 5-1Integrals of trigonometric functions$\displaystyle \int \sin x\,dx=-\cos x+C$6 examples · 6 patterns→
2. 5-2Integrals of exponential functions$\displaystyle \int e^x\,dx=e^x+C$6 examples · 5 patterns→
3. 5-3∫1/x dx = log|x| + C$\displaystyle \int \frac{1}{x}\,dx=\log|x|+C$6 examples · 6 patterns→
4. 5-4Substitution for (ax+b)$\displaystyle \int (ax+b)^{n}\,dx=\frac{1}{a}\cdot\frac{(ax+b)^{n+1}}{n+1}+C$6 examples · 2 patterns→
5. 5-5Integration by substitution$\displaystyle \int f(g(x))\,g'(x)\,dx=\int f(t)\,dt\quad(t=g(x))$6 examples · 4 patterns→
6. 5-6Partial fractions and integration$\displaystyle \frac{1}{(x-a)(x-b)}=\frac{1}{a-b}\left(\frac{1}{x-a}-\frac{1}{x-b}\right)\quad(a\neq b)$6 examples · 3 patterns→
7. 5-7Trigonometric substitution (x = a sin θ)$\displaystyle x=a\sin\theta\ \left(a>0,\ -\frac{\pi}{2}\le\theta\le\frac{\pi}{2}\right)$6 examples · 5 patterns→
8. 5-8Integration by parts$\displaystyle \int f(x)g'(x)\,dx=f(x)g(x)-\int f'(x)g(x)\,dx$6 examples · 6 patterns→
9. 5-9Substitution in definite integrals$\displaystyle \int_{a}^{b} f(x)\,dx=\int_{\alpha}^{\beta} f(g(t))\,g'(t)\,dt$6 examples · 3 patterns→
10. 5-10Integration by parts for definite integrals$\displaystyle \int_{a}^{b} uv'\,dx=\left[uv\right]_{a}^{b}-\int_{a}^{b} u'v\,dx$6 examples · 6 patterns→
11. 5-11Limits of Riemann sums$\displaystyle \lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n}f\!\left(\frac{k}{n}\right)=\int_0^1 f(x)\,dx$6 examples · 4 patterns→
12. 5-12Volume of a solid of revolution$\displaystyle V=\pi\int_a^b \{f(x)\}^2\,dx$6 examples · 3 patterns→
13. 5-13Arc length$\displaystyle L=\int_a^b\sqrt{1+\{f'(x)\}^2}\,dx$6 examples · 6 patterns→
14. 5-14Area under a parametric curve$\displaystyle S=\int_a^b y\,dx=\int_{\alpha}^{\beta} y\,\frac{dx}{dt}\,dt$6 examples · 6 patterns→

Updated 2026-10-16