Final Review 2: Applications
• Approximations:
√
– 4.04: f (x + ∆x) ≈ f (x) + f 0 (x) · ∆x where ∆x is very small compared to x.
– Newton’s Method: To solve the equation f (x) = 0, pick x1 that may be close to the solution,
f (xn )
then use the formula xn+1 = xn − 0
repeatedly.
f (xn )
– Numerical Integration: left/right/midpoint Riemann Sum.
• Sketching the graph:
– Increasing(decreasing): f 0 (x) > 0(< 0).
– Concave up(down): f 00 (x) > 0(< 0).
– (Global) Maximum and Minimum:
∗ Only works for continuous functions on CLOSED interval.
∗ Candidates are critical points: stationary points, endpoints and singular points.
∗ Evaluate the function at all critical points. The largest value(s) is the maximum and the
smallest value(s) is the minimum.
– Local Maximum and Minimum:
∗ Works for all continuous functions.
∗ A local extreme doesn’t have to be a global extreme.
∗ Candidates are critical points. Usually on open interval, hence no endpoint.
∗ First derivative test: If c is a critical point, test the sign change of f 0 (x) around c.
∗ Second derivative test: If c is a critical point, test the sign of f 00 (c).
– To determine the sign of f 0 (x), notice that the sign can only change at the critical points. In
the other word, between two adjacent critical points, the signs are all + or all −. Pick any
number a in each interval and test the sign of f 0 (a) will determine every sign in that interval.
• Differential Equations (Separating the Variable):
dy
into f 0 (x) and y into f (x) if necessary.
dx
2. Put f (x) and f 0 (x) terms on one side and x terms on the other side.
1. Rewrite
3. Take integral on both sides. Don’t forget the constants.
4. Simplify to the form f (x) =(a function in x) if possible.
• Length, Area and Volume: Figure out the setting of each formula.
Z b
– Area =
(f (x) − g(x)) dx
a
Z
b
π(f 2 (x) − g 2 (x)) dx
– Volume =
a
Z
b
x · (f (x) − g(x)) dx
– Volume = 2π
a
Z
b
– Length =
p
(f 0 (t))2 + (g 0 (t))2 dt
a
Z
– Area = 2π
a
b
p
g(t) (f 0 (t))2 + (g 0 (t))2 dt
– To find the implicit a and b, solve the equation f (x) = g(x).
– Sometimes you have to switch x and y.
• Practical Problems: Related rates, problems on extremes, work and center of mass.