Calculus practice by topic

No topics match. Try another word.

01Prerequisites: lines, exponents, logs, trig4 units

1. 1-1Equation of a line, y=a(x−p)+q$y=a(x-p)+q$6 examples · 6 patterns
2. 1-2Exponent rules and exponential functions$a^m a^n=a^{m+n},\quad (a^m)^n=a^{mn},\quad a^{-n}=\frac{1}{a^n},\quad a^{\frac{1}{n}}=\sqrt[n]{a}$6 examples · 4 patterns
3. 1-3Trigonometric functions and radians$180^\circ=\pi\ \text{rad},\qquad \ell=r\theta$6 examples · 5 patterns
4. 1-4Definition of the logarithm$\log_a b=c \iff a^c=b,\qquad \log_a xy=\log_a x+\log_a y$6 examples · 3 patterns

02Derivatives: the basics8 units

1. 2-1Difference quotient (and why you cannot divide by 0)$\frac{f(x+h)-f(x)}{h}\quad(h\neq 0)$6 examples · 5 patterns
2. 2-2The derivative at a point, f'(a)$f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$5 examples · 5 patterns
3. 2-3The derivative function f'(x)$f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$6 examples · 4 patterns
4. 2-4x^n from the definition$(x^n)'=nx^{n-1}$5 examples · 2 patterns
5. 2-5Power rule, constant multiples, sums and differences$(x^n)'=nx^{n-1},\quad (kf)'=kf',\quad (f\pm g)'=f'\pm g'$5 examples · 2 patterns
6. 2-6Equation of a tangent line$y=f'(a)(x-a)+f(a)$6 examples · 3 patterns
7. 2-7Sign of f' and increasing/decreasing$f'(x)>0 \Rightarrow \text{increasing},\quad f'(x)<0 \Rightarrow \text{decreasing}$6 examples · 3 patterns
8. 2-8Local maxima and minima$f'(a)=0,\ f':\ +\to -\ \Rightarrow\ \text{local max}\qquad f'(a)=0,\ f':\ -\to +\ \Rightarrow\ \text{local min}$6 examples · 3 patterns

03Integrals: the basics7 units

1. 3-1Antiderivatives (the reverse of differentiation)$F'(x)=f(x) \iff \int f(x)\,dx=F(x)+C$6 examples · 3 patterns
2. 3-2The formula for ∫x^n dx$\int x^n\,dx=\frac{x^{n+1}}{n+1}+C\quad(n\neq -1)$5 examples · 2 patterns
3. 3-3Constant multiples, sums and differences (polynomials)$\int kf\,dx=k\int f\,dx,\quad \int (f\pm g)\,dx=\int f\,dx\pm\int g\,dx$6 examples · 3 patterns
4. 3-4Area and antiderivatives (the FTC intuition)$S(x)=\int_0^x f(t)\,dt \;\Rightarrow\; S'(x)=f(x)$6 examples · 4 patterns
5. 3-5Definite integrals: F(b) − F(a)$\int_a^b f(x)\,dx=\Big[F(x)\Big]_a^b=F(b)-F(a)$6 examples · 5 patterns
6. 3-6Signed area$f(x)\le 0 \text{ on } [a,b] \Rightarrow \int_a^b f(x)\,dx\le 0$6 examples · 1 pattern
7. 3-7Area between two curves$S=\int_a^b \{f(x)-g(x)\}\,dx\quad(f(x)\ge g(x))$6 examples · 3 patterns

04Derivatives: rules and functions9 units

1. 4-1Derivatives of sin x and cos x$(\sin x)'=\cos x,\qquad (\cos x)'=-\sin x$6 examples · 6 patterns
2. 4-2Derivative of tan x$(\tan x)'=\frac{1}{\cos^2 x}$6 examples · 6 patterns
3. 4-3Derivative of e^x$(e^x)'=e^x,\qquad (a^x)'=a^x\log a$6 examples · 6 patterns
4. 4-4Derivative of log x$(\log x)'=\frac{1}{x},\qquad (\log|x|)'=\frac{1}{x}$6 examples · 6 patterns
5. 4-5Product rule$(fg)'=f'g+fg'$6 examples · 6 patterns
6. 4-6Quotient rule$\left(\frac{f}{g}\right)'=\frac{f'g-fg'}{g^2}$6 examples · 2 patterns
7. 4-7Meaning of a composite function f(g(x))$(f\circ g)(x)=f(g(x))$6 examples · 3 patterns
8. 4-8Chain rule$\{f(g(x))\}'=f'(g(x))\,g'(x)$6 examples · 6 patterns
9. 4-9Derivative of an inverse function$\frac{dy}{dx}=\frac{1}{\,\dfrac{dx}{dy}\,}$6 examples · 4 patterns

05Integrals: functions and by parts4 units

1. 5-1Integrals of trigonometric functions$\int \sin x\,dx=-\cos x+C,\qquad \int \cos x\,dx=\sin x+C$6 examples · 4 patterns
2. 5-2Integrals of exponential functions$\int e^x\,dx=e^x+C,\qquad \int a^x\,dx=\frac{a^x}{\log a}+C$6 examples · 5 patterns
3. 5-3∫1/x dx = log|x| + C$\int \frac{1}{x}\,dx=\log|x|+C$6 examples · 4 patterns
4. 5-4Integration by parts$\int f(x)g'(x)\,dx=f(x)g(x)-\int f'(x)g(x)\,dx$6 examples · 6 patterns

Updated 2026-09-05