Calculus I

I plan to have around 10 questions on the exam with at least one question from each section. You should be able to state any definition and any named theorem. You should be able to work any homework problem which has been assigned. Here is a list of important ideas:

• Everything you needed for exam 1. I won't ask any question specifically targeting exam 1 material, but you need to know this material in order to be able to answer questions from sections 3.1-3.10.
• Tangent and secant lines, how to find them.
• The definition of the derivative of a function at a point.
• The definition of the derivative as a function.
• Different notation for derivatives, ie Newton, Leibniz etc.
• Derivative rules (list not guaranteed to be comprehensive):
  • Derivative of a constant.
  • Derivative of \(x\).
  • Derivative of linear functions.
  • Power rule.
  • Linearity rules.
  • Derivative of \(e^x\).
  • Product rule.
  • Quotient rule.
  • Chain rule.
  • Derivative of trig functions.
  • Implicit differentiation.
  • Logarithmic differentiation.
  • Derivative of exponential functions.
  • Derivatives of hyperbolic trig functions.
• Differentiable implies continuous.
• AROC and IROC.
• Linear motion.
• Motion under in influence of gravity, position, velocity, acceleration.
• Higher derivatives, notation, etc.
• Related Rates

1. Let \(f(x)\) be a function and \(a\) a point.
   1. What does it mean for \(f(x)\) to be differentiable at \(a\)?
   2. Use the definition of derivative to find \(f'(a)\) for \(a=3\) and \(f(x) = 4-x^2\).
2. Find the equation of the line tangent to the graph of \(24e^x\) at \(x=2\).
3. Let \(f(x) = 5x+3\). Prove that \(\displaystyle \lim_{x \to 2} f(x) = 13\) using the formal definition of limit (that is the \(\epsilon, \delta\) definition).
4. Let \(\displaystyle g(s) = \frac{e^s}{s+1}\). Find \(g''(s)\).
5. The position of a particle is given by \(s(t) = t^2 -3t +4\).
   1. Find a formula for the velocity of the particle \(v(t)\).
   2. Find a formula for the acceleration of the particle \(a(t)\).
   3. At what time is the instantaneous velocity equal to the average velocity over the interval \([0,1]\).
6. Let \(V\) be the volume of a sphere of radius \(r\).
   1. Find \(\displaystyle \frac{dV}{dr}\).
   2. Find \(\displaystyle \frac{dV}{dr}\bigg|_{r=2}\).
7. Find the following derivatives:
   1. \(\displaystyle \frac{d}{dx} \left(\frac{ax+b}{cx+d}\right)\) for \(a,b,c,d\) constants.
   2. \(\frac{d}{dx}(2+x^{-1})(x^{3/2}+1)\).
   3. \(\frac{d}{dx} (7x^4-3x^2+x^{-1} + e^x +7\ln(x) +3^x)\).
   4. \(\frac{d}{d\theta} (\cos \theta + 7\tan \theta + \sec\theta)\).
   5. \(\frac{d}{dx} (4e^{-x} + 7x^{-2x})\).
   6. \(\frac{d}{dx} \cosh x\).
   7. \(\frac{d}{dx} \sqrt{1+\sqrt{x^2+1}}\).
8. Calculate \(\displaystyle \frac{dy}{dt}|_{(0,5\pi/2)}\) for \(y\cos(y+t+t^2) = t^3\).
9. Find \(\frac{dy}{dx}\) where \(3y^2+x^2 = 5\).
10. Find the derivative of \(\displaystyle f(x) = \frac{(x+1)^2(2x^2-3)}{\sqrt{x^2+1}}\). Hint: Use logarithmic differentiation.
11. Homework problem 3.10.7.