Solved examples: Finding functions from derivatives in calculus.

📚 Quick Study Guide

• 🔄 Antiderivatives: Finding a function given its derivative is called finding the antiderivative. Remember, if $F'(x) = f(x)$, then $F(x)$ is an antiderivative of $f(x)$.
• ➕ Constant of Integration: Since the derivative of a constant is zero, antiderivatives always have a '+ C', the constant of integration. So, the general antiderivative of $f(x)$ is $F(x) + C$.
• 📐 Power Rule for Integration: The integral of $x^n$ is $(\frac{x^{n+1}}{n+1}) + C$, where $n \neq -1$.
• 💡 Common Integrals:
  • The integral of $\sin(x)$ is $-\cos(x) + C$.
  • The integral of $\cos(x)$ is $\sin(x) + C$.
  • The integral of $e^x$ is $e^x + C$.
• 🎯 Initial Value Problems: You can find the specific antiderivative if you're given an initial condition, like $F(0) = 5$. Plug in the value to solve for C.

🧪 Practice Quiz

1. What is the antiderivative of $f(x) = 2x$?
   1. $x^2 + C$
   2. $2 + C$
   3. $x + C$
   4. $2x^2 + C$
2. What is the antiderivative of $f(x) = \cos(x)$?
   1. $\sin(x) + C$
   2. $-\sin(x) + C$
   3. $\tan(x) + C$
   4. $-\cos(x) + C$
3. What is the antiderivative of $f(x) = e^x$?
   1. $e^x + C$
   2. $xe^{x-1} + C$
   3. $e^{x+1} + C$
   4. $1 + C$
4. Find $F(x)$ if $F'(x) = 3x^2$ and $F(0) = 2$.
   1. $x^3 + 2$
   2. $x^3$
   3. $6x + 2$
   4. $x^3 + C$
5. What is the antiderivative of $f(x) = 5$?
   1. $5x + C$
   2. $5 + C$
   3. $0 + C$
   4. $x + C$
6. What is the antiderivative of $f(x) = x^3$?
   1. $\frac{x^4}{4} + C$
   2. $3x^2 + C$
   3. $x^4 + C$
   4. $\frac{x^2}{2} + C$
7. Find $F(x)$ if $F'(x) = 2x + 1$ and $F(1) = 4$.
   1. $x^2 + x + 2$
   2. $x^2 + x + C$
   3. $2x + 1 + 2$
   4. $x^2 + 2$