Calculus and Above

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1. find the first...

1. find the first and second derivatives for each of the functions. Locate any relative maximum or minimum points and any points of inflection. determine the intervals on which the function is concave upwards and concave downwards.

f(x)=√(x^2 +3)

2. Find f "(x). Then evaluate f" (0), f" (1), and f" (4) if they exist.

f(x)= (x)/(x+5)

3. Determine the intervals on which each function is a. concave upward, b. concave downward c. locate all points of inflection. Use the information gathered to to sketch the function.

f(x)= 1/3x^3-2x^2+x-3

f(x)= the cubed root of 2x+3

4. Sketch the graph of a continuous function that satisfies all the given conditions.

a. f(10)=5

b. f '(5)=0

c. f "(5)=10

d. f "(x)10

e. f "(x)>0 if x <10

5. Sketch the graph of each rational function. Show any related asymptotes on each graph.

f(x)=(x^3+x^2-13x+100) / (x-3)(x-10)

6. a front window on a new home is designed as a rectangle with a semicircle on the top. If the window is designed to let in a maximum amount of light, and the architect fixes the perimeter of the entire window at 600 inches, determine the radius r and rectangular height h so as to maximize the area.

f(x)=√(x^2 +3)

2. Find f "(x). Then evaluate f" (0), f" (1), and f" (4) if they exist.

f(x)= (x)/(x+5)

3. Determine the intervals on which each function is a. concave upward, b. concave downward c. locate all points of inflection. Use the information gathered to to sketch the function.

f(x)= 1/3x^3-2x^2+x-3

f(x)= the cubed root of 2x+3

4. Sketch the graph of a continuous function that satisfies all the given conditions.

a. f(10)=5

b. f '(5)=0

c. f "(5)=10

d. f "(x)10

e. f "(x)>0 if x <10

5. Sketch the graph of each rational function. Show any related asymptotes on each graph.

f(x)=(x^3+x^2-13x+100) / (x-3)(x-10)

6. a front window on a new home is designed as a rectangle with a semicircle on the top. If the window is designed to let in a maximum amount of light, and the architect fixes the perimeter of the entire window at 600 inches, determine the radius r and rectangular height h so as to maximize the area.

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