Chapter 3 Test Review

Hello Calculus students! Here is your review packet to help you prepare for the upcoming Chapter 3 test. This chapter focuses heavily on derivatives, so be sure to practice, practice, practice! Remember to utilize your notes and examples from class as you work through these problems.

Here's a breakdown of what you can expect to see on the test and in this review:

Key Concepts

  • Basic Derivative Rules: Power Rule, Constant Multiple Rule, Sum/Difference Rule. Remember the power rule: if $f(x) = x^n$, then $f'(x) = nx^{n-1}$.
  • Derivatives of Trigonometric Functions: Know your derivatives for $\sin(x)$, $\cos(x)$, $\tan(x)$, $\csc(x)$, $\sec(x)$, and $\cot(x)$. For example, the derivative of $\sin(x)$ is $\cos(x)$, and the derivative of $\cos(x)$ is $-\sin(x)$.
  • Exponential and Logarithmic Functions: Derivatives of $e^x$ and $\ln(x)$ are essential. The derivative of $e^x$ is itself ($e^x$), and the derivative of $\ln(x)$ is $\frac{1}{x}$. Don't forget that the derivative of $b^x$ is $b^x \ln(b)$.
  • Product Rule: If $f(x) = u(x)v(x)$, then $f'(x) = u'(x)v(x) + u(x)v'(x)$.
  • Quotient Rule: If $f(x) = \frac{u(x)}{v(x)}$, then $f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$.
  • Chain Rule: If $f(x) = g(h(x))$, then $f'(x) = g'(h(x)) \cdot h'(x)$. This is crucial for composite functions!
  • Implicit Differentiation: Used when you can't explicitly solve for $y$ in terms of $x$. Remember to use the chain rule whenever you differentiate a term involving $y$.
  • Linear Approximation: Using the tangent line to approximate function values. The linear approximation of $f(x)$ at $x=a$ is given by $L(x) = f(a) + f'(a)(x-a)$.

Practice Problems

Here are some practice problems similar to what you'll see on the test. Work through them carefully, showing all your steps. Check your answers against the answer sheet provided.

  1. Find the derivative of $f(x) = \frac{x^2 - x + 2}{\sqrt{x}}$.
  2. Calculate the derivative of $f(x) = (x^2 + 3)^2$.
  3. Determine the derivative of $f(x) = x^4 + 3x^{3/2} - 5 - 2x^{-2}$.
  4. Find the derivative of $f(x) = (3x^2 - 5x)e^x$.
  5. Use implicit differentiation to find $\frac{dy}{dx}$ for the equation $x^2 - 4xy + y^2 = 4$.
  6. Find the linear approximation of $f(x) = \sqrt{1-x}$ at $a=0$.

Remember: Understanding the underlying concepts is more important than just memorizing formulas. Good luck with your studying!

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