Practice Sheet for Natural Logarithmic and
Natural Exponential Functions
I. Find the formula for the inverse function for each of the following functions.
Also, find the domains of and .
(1) (2)
(3) (4)
(5) (6)
II. Differentiate the following functions:
(1) (2)
(3) (4)
(5) (6)
(7) (8)
(9) (10)
(11) (12)
(13) (14)
(15) (16)
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(17) (18)
(19) (20)
(21) (22)
III. Evaluate the following integrals:
(1) (2)
(3) (4)
(5) (6)
(7) (8)
(9) (10)
(11) (12)
(13) (14)
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Solution Key for Natural Logarithmic and
Natural Exponential Functions
I. (1) To get the domain of f , set . Thus, the domain of f is
. To find the formula for, use the 3-step process:
(i)
(ii)
(iii) Solve for y: .
Thus, , and the domain of is all real numbers.
(2) The domain of f is all real numbers. To find the formula for , use the
3-step process:
(i)
(ii)
(iii) Solve for y: .
Thus, , and the domain of is .
(3) To get the domain of f , set . Thus, the domain of f is
. To find the formula for, use the 3-step process:
(i)
(ii)
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(iii) Solve for y:
. Thus,
, and the domain of is all real numbers.
(4) The domain of f is all real numbers. To find the formula for , use the
3-step process:
(i)
(ii)
(iii) Solve for y:
. Thus,
. To get the domain of , set
domain of = .
(5) The domain of is and the domain of is , so the
domain of f is . To find the formula for, use the 3-step
process:
(i)
(ii)
(iii) Solve for y:
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. Thus,
, and the domain is .
(6) Obviously, the domain of f is . To find the formula for, use the
3-step process:
(i)
(ii)
(iii) Solve for y: using the
quadratic formula, . In order to
determine which of the two formulas is the correct one, go back to the original
function and substitute .
and .
Thus, . To get the domain of , set
or . But, > 0 for all x domain of =
.
II. (1)
(2)
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(3) Use the product rule
(4) Use the product rule
(5) Use the quotient rule
(6) Use the quotient rule
(7)
(8)
(9)
(10)
(11) Simplify using the laws of logarithms first before you differentiate
(12) Simplify using the laws of exponents first before you differentiate
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(13) Simplify using the laws of logarithms first before you differentiate
(14)
(15)
(16)
(17)
(18)
(19)
(20) Use logarithmic differentiation:
(i)
(ii)
(iii)
(iv)
(v)
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(21) Use logarithmic differentiation:
(i)
(ii)
(iii)
(iv)
(v)
(22) Use logarithmic differentiation:
(i)
(ii)
(iii)
(iv)
(v)
III. (1) Let
(2) Let
(3) Let
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(4) Let
(5) Let
(6) Let
(7) Let
(8)
(9) Let
(10)
(11) Let
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(12) Let
(13) Let
(14) let
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