18.01 Section, October 21, 2015
Section leader: Eva Belmont (ebelmont@mit.edu, E18-401B)
Midterm 2 review
Things you are supposed to memorize
• Taylor approximations:
x2 x3 x4
+
+
+ ...
2!
3!
4!
x3 x5 x7
sin x ≈ x −
+
−
+ ...
3!
5!
7!
x2 x4 x6
+
−
+ ...
cos x ≈ 1 −
2!
4!
6!
x3 x5 x7
arctan x ≈ x −
+
−
+ ...
3
5
7
1
1
1
ln x ≈ (x − 1) − (x − 1)2 + (x − 1)3 − (x − 1)4 + . . .
2
3
4
• Differential equations:
◦ See the differential equations above
ex ≈ 1 + x +
◦ Solutions to
conditions)
dy
dt
at a = 0
at a = 0
at a = 0
at a = 0
at a = 1
= c have the form y(t) = ct + b (for some constant b, usually determined by initial
2
◦ Solutions to ddt2y = c have the form y(t) = 2c t2 + bt + a (for some constants a, b, usually determined by
initial conditions)
◦ Solutions to
d2 y
dt2
◦ Solutions to
d2 y
dt2
= −cy (where c > 0) have the form
√
√
y(t) = A cos( c · t) + B sin( c · t)
= cy (where c > 0) have the form
√
y(t) = Ae
or, with different constants,
c·t
√
+ Be−
c·t
√
√
y(t) = C sinh( c · t) + D cosh( c · t)
Review
• Limits
◦ How to compute limits:
x
n
n
I Estimation via order-of-magnitude calculations: as x → ∞, e grows faster than x , and x grows
1
faster than ln x. As x → 0, xn grows faster than −ln x.
∗ Technique for comparing growth of f (x) vs. g(x) as x → ∞: how does the increase from f (x)
to f (2x) compare to the increase from g(x) to g(2x)?
f (x)
f 0 (x)
= lim 0
(if the latter limit exists)
x→a g(x)
x→a g (x)
sin x
x
I Using Taylor series (e.g. lim
≈ lim = 1)
x→0 x
x→0 x
◦ Intermediate value theorem
I Input: a function f , and two numbers a, b such that f (a) < 0 and f (b) > 0
I
L’Hôpital’s rule: if f (a) = 0 = g(a), then lim
I
Output: (existence of) a zero of f
◦ Intersection problems using IVT (f (x) = g(x), fixed points, . . . )
◦ Interval bisection method for approximating roots of a function
• Taylor polynomials
1
◦ nth order Taylor approximation for f at a is:
1
1
f (x) ≈ f (a) + f 0 (a)(x − a) + f 00 (a)(x − a)2 + · · · + f (n) (a)(x − a)n
2!
n!
◦ Know the specific ones above, and know how to use this formula to compute others from scratch
◦ Be on the lookout for situations that require an approximation
• Piecewise function problems (e.g. review problem 6 on pset 5)
• Differential equations and “physics problems”
◦ See the differential equations above
◦ Look at section 9.6 in the book and go through the worked examples in the text about pendulums and
springs.
√
√
2π
◦ Period of y(t) = A cos( c · t) + B sin( c · t) is √
c
◦ Hyperbolic sin and cosine
cosh(x) =
ex + e−x
2
sinh(x) =
d
cosh x = sinh x
dx
ex − e−x
2
d
sinh x = cosh x
dx
• Convexity
◦ Second derivative test
f 00 (t) > 0 for t in [a, b]
⇐⇒
f is concave up in [a, b]
⇐⇒
there is at most one critical point in [a, b]
and (if it exists) it is a local min
f 00 (t) < 0 for t in [a, b]
⇐⇒
f is concave down in [a, b]
⇐⇒
there is at most one critical point in [a, b]
and (if it exists) it is a local max
◦ How to remember this? Think about x2 and −x2 .
◦ Should know how to find and classify critical points (use in the region of the critical point to say whether
it’s a local min, max, or neither)
Problems
1. (a) Use order-of-magnitude estimation to guess the limit lim
x→∞
(b) Evaluate lim
x→0
2x
.
ex
2x
using L’Hôpital’s rule.
−1
ex
(c) Evaluate the limit in part (b) using a Taylor approximation.
2
2. Show that cos x has a fixed point.
3. Find the value a (if any) that makes the following function continuous:
(
x
if x > 0
f (x) = ln(1+x)
a
if x ≤ 0.
4. Find the cubic p(x) that makes the following function continuous, with continuous derivative:


if x < 0
0
f (x) = p(x) if 0 ≤ x < 1


−1
if x > 1.
5. A spring is attached to the wall, and a weight is attached to the other end of the spring. The weight is
pulled 0.1 m from the equilibrium position, and then let go. Suppose the period is 2 seconds. What is the
weight’s position after 0.25 seconds?
6. Find and classify (using the second derivative test) the critical points of x3 − 3x.
3
Problems to think about at home
1. Use the material on this midterm to come up with as many ways as you can to approximate ln 2 without
using a calculator.
I can think of at least three: (1) using a Taylor approximation for ln x; (2) using a second-order Taylor approximation for ex ;
(3) using a third-order Taylor approximation for ex along with the interval bisection method.
2. (a) Find the polynomial p(x) of minimal degree that makes
(
sin(x) if x < 0
f (x) =
p(x)
if x ≥ 0
continuous, with continuous first derivative.
(b) What if you also require the second derivative to be continuous?
(c) What if you require all derivatives up to f (k) to be continuous?
3. Using only the Taylor approximations on the “you should memorize this” list, come up with fourth order
Taylor approximations for the following:
(a) e−x at a = 0
(b) x sin x at a = 0
(c)
1
x2 +1
(d)
1
x+1
at a = 0
at a = 0
4. Approximate
√
3 using interval bisection, to an error of at most 18 .
4