Download Calculus I Midterm II Review Materials Solutions to the practice

Calculus I Midterm II Review Materials
Solutions to the practice problems are provided in another sheet, it’s recommended that you first do the problems by yourself and then check with the
answer.
1, Definition of Derivative, Differentials? Relation with change rate, tangent,
velocity? Express a limit as √
a derivative and evaluate.
4
sin x2 − sin 9
16 + h − 2
e.g. lim
, lim
.
x→3
h→0
x−3
h
2,Ways to calculate derivative.
I, Standard functions:
Memorize the derivative of C, xa , ex , lnx , sin x, cos x, tan x. Know how to use
definition to get derivative of xn , where n is a positive integer.
II, Derivative Laws for (+, −, ×, ÷) and scalar multiplication cf (x).
(f + g)0 = f 0 + g 0 , (f − g)0 = f 0 − g 0 , (cf )0 = cf 0
0
g0
(f g)0 = f 0 g + f g 0 , ( fg )0 = f g−f
g2
2x
x
1+t
1
, y = cos
e.g. f (x) = √
3 4 + 3x − x ln
ex , f (t) = 1−t2
x
III, Chain rule for f ◦ g: (f ◦ g)0 (x) = f 0 (g(x))g 0 (x).
e.g. e1/x , sin 3x, cos(tan x)
IV, Combination of II and III.
2
√
e.g. e−x cos x, tan2 (sin 2θ), ln3x +2x−1 ,
Find f 0 in terms of g 0 if f (x) = x2 g 2 (x).
V, ln method for f (x)g (x). The formula is
(f g )0 = f g (g ln f )0 .
x
−x
e.g. 3x ln , xx , sin xe .
VI, When y is given by an implicit equation: Derive the equation on both sides,
use techniques from I to V to calculate derivatives on both sides and get an
equation with only x, y, y 0 (in this process consider y as a function of x, x as the
variable, unless specified otherwise as in the quiz), finally solve the equation for
y0 .
e.g. sin(xy) = x2 − y
e.g. x2 + y 2 = 25, x, y are functions of t, if
3, Higher derivatives
e.g. y =
√
dx
dt
= 2x, find
4x + 1, find y 00 .
e.g. y = cos 2x, find y (30)
4,Tangent line to a curve y = f (x) at x = x0 :
y − y0 = f 0 (x0 )(x − x0 )
1
dy
dt
when x = 3.
a, it’s f 0 (x0 ), but not f 0 (x).
b, y0 = f (x0 ).
e.g. y = 4 sin2 x when x = π/6,
e.g. At what points on the curve y = cos x+sin x, 0 ≤ x ≤ 2π, is the tangent
line horizontal?
5, Linear Approximation of f (x) near x = x0 .
f (x) ≈ f (x0 ) + f 0 (x0 )(x − x0 )
(Make sure get every term correct in the plugging in process. It is the same as
the tangent line formula!(Do the comparison by yourself))
√
e.g. Find the linear approximation of f (x) = 25 − x2 near 3, use it to
estimate f (3.01).
e.g. Use linear approximation to estimate e−0.015 .
6, Maximum and Minimum for f (x) on a closed interval [a, b].
Step1: Verify f (x) is continuous.
Step2: Find critical values by solving f 0 (c) = 0.
Step3: Find endpoint values f (a), f (b).
Step4: The largest value from step2 and step3 is absolute max., and the
smallest absolute min..
e.g. find absolute max. and min. of f (x) = 3x2 − 12x + 5 on [0, 3].
2