# Derivatives and continuity of functions

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.

Thanks for asking BrainMass.

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.

Calculus Problems : Continuity, Limits, Derivatives, Inequalities, Quadratic Equations and Parabolas and Maximum Height

1. Solve log₆x-3=0

2. A business owner is comparing the costs of purchasing inventory and the profit from the sale of the product. The relationship proves to be linear. Which type of variation will describe the data?

Direct as nth power joint regress

Inverse direct

3. Solve x²-25<0

4. What is the limit as x approaches -3 for the function f(x)=x³

5. Determine points of discontinuity, if any, for the function f(x)= 1/x+1 on the interval [-10, 10]

6. What is the derivative of the function f(x)= 3x² + 2x -5

7. f(x) = 2x² - 3x + 2: Approximate derivative of f(x) to nearest tenth when x=2.71

8. f(x) = 6x + 2; Approximate an anti- derivative of f(x) to the nearest tenth when x = 3.4

9. Height of an object shot straight upward from the ground with an initial velocity of 64ft per square is given by the equation h = 64t - 16t². What is the maximum height reached by the object?

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