Exam 2 Review

Exam 2 Review

1. Intermediate Value Theorem

   • Conditions for IVT
     • f is continuous
     • [a,b] is closed interval
     • f(a) can't equal f(b)
   • For every value M between f(a) and f(b), there exists one value c within (a,b) such that f(c) = M
     • Basically, a continuous function can't skip values
   • IVT is false for functions defined only on the rational numbers

2. Bisection Method

   • Helps to find zeroes by bisecting the interval they must exist on

3. Motion Under the Influence of Gravity

   • s(t) = s0 + v0t - 1/2gt^2
   • v(t) = ds/dt = v0 - gt

4. Linear Approxmation

   • ▲f ≈ f'(a)(▲x)
     • f'(a) ≈ ▲f/▲x
   • Uses the the derivative to estimate the change in f without calculating it exactly

5. Linearization

   • Calculates the function f itself (not ▲f)
   • L(x) = f'(a)(x-a) + f(a)

6. Implicit Differentiation

   • 1. Isolate y 2. Solve for dy/dx

7. Explicit Differentiation

   • 1. Take the derivative of both sides 2. Isolate dy/dx

8. Definitions

   • Derivative at a point - slope of secant line at that point
   • Derivative of a function - asking for a formula (for the slope of the tangent line)
   • Local min - min in a certain interval
   • Local max - max in a certain interval
   • Critical point - where the derivative = 0
     • Finding critical points
       • 1. Take the derivative 2. Set derivative = 0 3. Solve for x
   • Inflection point - point where concavity changes and f''(x) = 0

9. Mean Value Theorem

   • Conditions for MVT
     • f is continuous
     • [a,b] is a closed interval
     • f is differentiable on (a, b)
   • f'(c) = (f(b) - f(a))/(b - a)
   • The case of Rolle's Theorem
     • f(a) = f(b), f'(c) = 0

10. Fermat's Theorem

    • If f'(c) is a local max/min, then c is a critical point of f(x)

11. First Derivative Test

    • 1. Find critical points 2. Pick test points If f'(tstpt) > 0, f is inc. If f'(tstpt) < 0, f is dec. If f goes from inc to dec, cp = max. If f goes from dec to inc, cp = min.
    • Finds mins/maxes

12. Review notes

    • Trig Function Derivatives
    • Inverse Trig Function Derivatives
    • Log/Natural Log Derivates
    • Related Rates

13. 2nd Derivative Test

    • 1. Find critical points 2. Find f'' 3. f''(cp) If f''(cp) > 0, f(c) = local max. If f''(cp) < 0, f(c) is a local min. If f''(cp) = 0, test is inconclusive.
    • Finds mins/maxes and concavity
    • Doesn't always work

14. L'Hôpital's Rule

    • If lim as x → ∞ of f/g = 0/0 or ∞/∞, then lim x → ∞ of f/g = lim x → ∞ f'/g'