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Using calculus, determine any intervals where the function below would be concave upward. If there are none, write ‘none’ in the answer box. (8 points) f ( ) x x^4^ 8 x^3^ 18 x^2 8
For which value of x of the function, f ( ) x x^4 8 x^3^ 18 x^2 8 (problem 1 function), is the function both decreasing and concave down? Select the correct choice. (8 points)
A x B x C x D x E x
Interval(s) of Concave Upward:
- Find the equation(s) of any vertical or horizontal asymptotes for the function below. Write ‘none’ if there are no asymptotes for a category. If there if more than one for a category, separate equations with commas. (6 points) 2 2
x x
r x
x x
- Using calculus, find any ordered pair(s) (point(s)), where the function below has a relative maximum or a relative minimum. Write as ordered pair(s) or point(s). (10 points) 2 ( ) 2
f x x x
Vertical Asymptote Equation(s):
Horizontal Asymptote Equation(s);
Relative maximum(s):
Relative minimum(s):
Find the derivative of y 2 x e^3 x. (9 points)
Solve this equation, using algebra/calculus. log 4 x log ( 4 x 3) 1 (8 points)
x =
- Graph the following function, using the information and points (1 – 5) shown below. (8 points) f ( ) x x^3^ 6 x^2 9 x
- y- intercept: (0, 0)
- Increasing: (^) ( , 3) and ( 1, ); decreasing( 3, 1)
- Relative minimum at ( 1, 4)and relative maximum at( 3,0)
- Point of inflection: ( 2, 2)
- Concave downward: ( , 2), concave upward: ( 2, )
- Which of the statements in the box is(are) true concerning the function f ( ) x x x ( 3)^2? Support your answer with work or an explanation. (10 points)
A I, II, III, and IV B II and IV only C I and IV only D I and II only E I, II, and III only
I The function f is decreasing on (1, 3). II There is a point of inflection at (2, 2). III The function f is concave upward on (2, ∞). IV There is a relative maximum.
