MATH 18.01 - MIDTERM 1 REVIEW: SUMMARY OF SOME KEY
CONCEPTS (WARNING: THERE MAY BE TYPOS, SO YOU SHOULD
CHECK EVERYTHING ON YOUR OWN!!)
18.01 Calculus, Fall 2014
Professor: Jared Speck
a. Ways of thinking about derivatives
(a) Analytic definition:
f (x + ∆x) − f (x)
∆x→0
∆x
(b) Geometric interpretation: f 0 (x0 ) = slope of tangent line to the graph of f at (x0 , f (x0 ))
dy
(c) dx
= instantaneous rate of change of y with respect to x
b. Tangent lines:
(a) y − f (x0 ) = f 0 (x0 )(x − x0 )
c. Derivative rules (know how to prove them)
(a) Sum: (u + v)0 = u0 + v 0
(b) Constant multiple: (cu)0 = cu0 if c is a constant
(c) Product: (uv)0 = uv 0 + u0 v
(d) Quotient: (u/v)0 = (u0 v − uv 0 )/v 2
(e) Chain:
f 0 (x) = lim
d
f (g(x)) = f 0 (g(x))g 0 (x),
dx
dy
dy du
=
dx
du dx
d. Limits including how to deduce them (here are some important examples)
(a) limθ→0 sinθ θ = 1
θ
(b) limθ→0 1−cos
=0
θ
cos θ−1
(c) limθ→0 θ2 = − 21
(d) limk→∞ (1 + k1 )k = e
e. Continuity
(a) Analytic definition: lim∆x →0 f (x + ∆x) = f (x)
(b) Jump discontinuities
(c) Removable discontinuities
(d) Discontinuities that are neither jumps nor removable
(e) Differentiable =⇒ continuous (know how to prove this)
1
Midterm 1 - Review Sheet
2
f. Derivatives of elementary functions including how to deduce the formulas (here are some
examples):
d
(a) dx
sin x = cos x
d
(b) dx cos x = − sin x
d
tan x = sec2 x
(c) dx
d r
x = rxr−1
(d) dx
d x
e = ex
(e) dx
d x
(f) dx
a = (ln a)ax
d
(g) dx
ln x = x1
d
1
(h) dx arcsin x = √1−x
2
d
1
(i) dx arccos x = − √1−x2
1
d
arctan x = 1+x
(j) dx
2
d
(k) dx
sinh x = cosh x
d
(l) dx
cosh x = sinh x
g. Function inverses
(a) f (f −1 (x)) = x
(b) f −1 (f (x)) = x
d −1
(c) If y = f (x) and x = f −1 (y), then dy
f (y) = d f1(x) = f 01(x)
dx
(d) The graph of f −1 is the reflection of the graph of f through the line y = x
(e) Example: ln x and ex are inverses of each other
h. Logarithmic differentiation
(a) Main point: if y = f (x), then sometimes ln y is easier to differentiate than y
0
d
(b) If y = f (x), then dx
ln y = yy