MATH 1431 HCCS Inverses Exponentials and Logarithms Questions

Math 1431Homework Assignment 10 (Written)
Inverses, Exponentials, and Logarithms (4.1 – 4.4)
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1. (10 points) The equation of the line tangent to the differentiable and invertible function f (x) at the point (5, 12)
is given by y = −3x + 27. Find the equation of the tangent line to f −1 (x) at the point (12, 5).
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2. (5 points) Several statements about a differentiable, invertible function f (x) and its inverse f −1 (x) are written
below. Mark each statement as either “TRUE” or “FALSE” (no work need be included for this question).
1. If f (π) = 2022 then π = f −1 (2022).
2. If f is increasing on its domain, then f −1 is also increasing on its domain.
3. The domain of f −1 is the range of f .


4. If x is in the domain of f −1 then f f −1 (x) = x.
5. If f is concave up on its domain then f −1 is concave up on its domain. (Hint: think about the examples f (x) = ex and
f −1 (x) = ln x.)


3. (10 points) Determine where the function f (x) = ln x2 + 1 is increasing and decreasing.
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4. (10 points) Shasta the Cougar was working at differentiating a function p(x) and they found that p0 (0) = ln
Unfortunately, Shasta forgot the exact formula for the original function p(x). They can only remember that
p(x) = xa − ax
for some constant a > 1. Help out our four-legged mascot-mathematician by determining the value of the constant a.

3
5. (12 points) (a) (8 points) Use logarithmic differentiation to find the derivative of y = x2 + 1 (x − 1)6 x2

3
(b) (4 points) Find the equation of the line tangent to the graph of y = x2 + 1 (x − 1)6 x2 at the point (2, 500).
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1
.
π
6. (10 points) Five equations are written below. Fill in the missing expression for each equation (2 points each and
no work need be shown for this problem).
0



ex
loga
0
0

0

0
7. (10 points) Use logarithmic differentiation to compute
= ax ln a.
=
.
=
.
=
1
.
x
=
1
1 + x2

(2 + sin x)x
4
0
.
√


8. (10 points) Find the equation of the line tangent to the graph of y = tan sin−1 x when x = 2/2.
9. (13 points) Consider the function f (x) = arctan
√

4 − x2 .
(a) (3 points) What is the domain of f (x)?
(b) (10 points) On what intervals is f (x) increasing? On what intervals is f (x) deceasing?
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10. (10 points) As we will see in Section 4.5, two “new” functions are defined in terms of the exponential functions
ex and e−x , namely
cosh(x) =
ex + e−x
2
sinh(x) =
ex − e−x
2
Use our rules for differentiating ex to show that
cosh0 (x) = sinh(x)
sinh0 (x) = cosh(x)
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