University:
Amherst College
Course:
MATH 121 | Intermediate Calculus
Academic year:

2023
Views:

416

Pages:

2
Author:

Arthur Roberson

Overview of ex and ln x Functions • Graphs • Function Properties Natural Exponential Function y = ex Natural Logarithm Function y = ln x • Domain = R • Domain = (0, ∞) • Range = (0, ∞) • Range = R • lim ex = ∞ • lim ln x = ∞ x→∞ • x→∞ lim ex = 0 • lim ln x = −∞ x→−∞ x→0+ eh − 1 =1 h→0 h • e chosen so that lim • y = ln x is the inverse function for y = ex • 2 < e < 3 and e ≈ 2.71828 . . . • y = ln x if and only if ey = x  x  ln e = x for all x in R ¶ µ 1 n • e = lim 1 + n→∞ n • • e0 = 1 and e1 = e • ln 1 = 0 and  ln x e = x for all x > 0 ln e = 1 • Algebraic Properties Natural Exponential Function y = ex Natural Logarithm Function y = ln x • ex ey = ex+y • ex−y ex = y e • ln(xy) = ln x + ln y µ ¶ x • ln = ln x − ln y y • (ex )y = exy • ln(xy ) = y ln x • e(x ln(x + y) or y) cannot be simplified ln x cannot be simplified ln y • Derivatives and Integrals Natural Exponential Function y = ex Natural Logarithm Function y = ln x • d x e = ex dx du d u(x) e = eu(x) = eu(x) u′ (x) dx dx Z • ex dx = ex + C • • d 1 ln x = dx x d 1 du u′ (x) ln u(x) = = dx u(x) dx u(x) Z 1 • dx = ln |x| + C x •

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