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# Derivatives and continuity of functions

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1. Find the derivative of the function.
h(t) = t2(3t + 9)3

2. Find the derivative of the function.

3. Find the derivative of the function.

Find the derivative of the function.
4. f(x) =
ex + e−x
------------
2
Find the derivative of the function.
5. f(x) =
3 ln x
---------------
x4

6.
Find dy/du, du/dx, and dy/dx when y and u are defined as follows.
y = √u u = 5x - 7x2
dy/du =

du/dx=

dy/dx=

7. Find an equation of the tangent line to the graph of the function at the given point.

y =

8. The population of Americans age 55 and older as a percentage of the total population is approximated by the function
f(t) = 10.72(0.9t + 10)0.3 (0 ≤ t ≤ 20)
where t is measured in years, with t = 0 corresponding to the year 2000. At what rate was the percentage of Americans age 55 and older changing at the beginning of 2006? (Round your answer to four decimal places.)
% per year =

At what rate will the percentage of Americans age 55 and older be changing in 2019? (Round your answer to four decimal places.)
% per year =

What will be the percentage of the population of Americans age 55 and older in 2019? (Round your answer to two decimal places.)
% per year =
9. Use the graph of the function f to find the limits at the indicated value of a, if the limit exists.

10. Find the indicated one-sided limits, if they exist.
f(x) =
−x + 2 if x ≤ 0
3x + 4 if x > 0

−x + 2 if x ≤ 0
3x + 4 if x > 0
lim x→0+ f(x) =
lim x→0− f(x) =

11. Determine all values of x at which the function is discontinuous.

x = (smaller value)
x = (larger value)

12. For what value of k will the function f be continuous on (-∞,∞)?

13. Determine all values of x at which the function is discontinuous.

x = (smaller value)
x = (larger value)

14. Determine the values of x, if any, at which the function is discontinuous. At each number where f is discontinuous, state the condition(s) for continuity that are violated. (Select all that apply.)

Which of these are true? Select all that apply.

The function of x is discontinuous at x = -1 (because) lim_(x -1)f(x) (exists, but this limit is not equal to) f(-1).

15. The following graph shows the amount of home heating oil remaining in a 200-gal tank over a 120-day period (t = 0 corresponds to October 1). For which value(s) of t is the function discontinuous. (Enter your answers as a comma-separated list.)
t =

https://brainmass.com/math/basic-calculus/derivatives-continuity-functions-561572

#### Solution Preview

Hi there,

Thanks for letting me work on your post. I've included my explanations in the word document.

1. Find the derivative of the function.
h(t) = t2(3t + 9)3

2. Find the derivative of the function.

3. Find the derivative of the function.

Find the derivative of the function.
4. f(x) =
ex + e−x
------------
2
Find the derivative of the function.
5. f(x) =
3 ln x
---------------
x4

6.
Find dy/du, du/dx, and dy/dx when y and u are defined as follows.
y = √u u = ...

#### Solution Summary

The derivatives and continuity of functions are examined.

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