University of California Irvine Math Algebra Worksheet

Unit 4 Mega-assignmentSection 4.1
1. What is the balance after 18 months in an account where $1000 was invested at an annual
interest rate of 9%, compounded monthly? First, give an exact expression and then use a
calculator or spreadsheet to get the amount of money to the nearest penny.
Section 4.2
2. Calculate the limits:
ˆ
˙k
2
(a) lim 1 `
kÑ8
k
ˆ
˙k
2
(b) lim 1 ´
kÑ8
k
ˆ
˙k
lnp2q
(c) lim 1 `
kÑ8
k
3. What is the balance after 18 months in an account where $1000 was invested at an annual
interest rate of 9%, compounded continuously? First, give an exact expression and then use
a calculator or spreadsheet to get the amount of money to the nearest penny.
4. How long does it take for the investment in problem 3 to double to $2000? . . . to triple
to $3000? . . . to become $100,000?
Section 4.3
5. By thinking about the graphs of the relevant exponential functions, calculate
(a) lim 2x
xÑ8
(b) lim 2x
xÑ´8
ˆ ˙t
1
.
(c) lim 3 , which is equal to lim
tÑ8
tÑ8
3
´t
(d) lim 3´t .
tÑ´8
6. What is the balance on a 30-year mortgage of $100,000 at an annual interest rate of 5%
after
(a) 2 years?
(b) 10 years?
(c) 25 years?
(d) What is the monthly (principal and interest) payment on the mortgage?
For each part, provide both the relevant expression and the value to the nearest penny.
Section 4.4
7. A bacterial colony is growing exponentially. It starts with 4000 bacteria and after 3 days
there are 6000. How many bacteria will there be after 15 days?
8. A radioactive substance has a half-life of 15 years. How long will it be until only 1 gram
of a 64-gram sample of the substance remains?
9. In a room where the temperature is 70 degrees, the temperature of a cup of coffee
decreases from 198 degrees to 134 degrees in 2 minutes. In how many more minutes will the
temperature of the coffee be 86 degrees?
Section 4.5
10. Calculate the derivatives of:
(a) f pxq “ 3x
(b) gpxq “ x6
(c) hpxq “ lnpxq
11. What is the equation of the tangent line to the graph of y “ x3 at the point p2, 8q?
(The derivative tells you the slope, right?)
12. What is the relative rate of change of f pxq “ 3x ? . . . of gpxq “ x3 ?
Section 4.6
13. In a dataset with 500 numbers in it, about how many of them would you expect to begin
with the digit 8, based on Benford’s law?
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