APPLIED CALCULUS FOR LIFE SCI

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Student: _____________________ Date: _____________________

Instructor: Edward Tawiah Course: Applied Calc Assignment: Test 3

For the function below, find a) the critical numbers; b) the open intervals where the function is increasing; and c) the open intervals where it is decreasing.

f(x) = (x + 3) 2 / 3

a) Select the correct choice below and fill in any answer boxes in your choice.

A. The critical number(s) is/are . (Type an integer or a simplified fraction. Use a comma to separate answers as needed.)

B. There is no critical number.

b) Select the correct choice below and fill in any answer boxes in your choice.

A. List any interval(s) on which the function is increasing.

(Type your answer in interval notation. Simplify your answer. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.)

B. The function is never increasing.

c) Select the correct choice below and fill in any answer boxes in your choice.

A. List any interval(s) on which the function is decreasing.

(Type your answer in interval notation. Simplify your answer. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.)

B. The function is never decreasing.

Find the x-value of all points where the function below has any relative extrema. Find the value(s) of any relative extrema.

f(x) = − x − x − 32

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A. The function has a relative at the point(s) .maximum (Simplify your answer. Type an ordered pair. Use a comma to separate answers as needed.)

B. The function has no relative .maximum

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A. The function has a relative at the point(s) .minimum (Simplify your answer. Type an ordered pair. Use a comma to separate answers as needed.)

B. The function has no relative .minimum

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After a great deal of experimentation, two college senior physics majors determined that when a bottle of French champagne is shaken several times, held upright, and uncorked, its cork travels according to the function below, where s is its height (in feet) above the ground t seconds after being released.

s(t) = − 16t + 60t + 42

a. How high will it go? b. How long is it in the air?

a. The cork will go feet. (Round to the nearest whole number as needed.)

b. The cork will be in the air for seconds. (Type an integer or decimal rounded to two decimal places as needed.)

For the function f(x) , find f (x). Then find f ( ) and f ( ).= x2 + 4 ′′ ′′ 0 ′′ 6

f (x) ′′ =

Select the correct choice below and fill in any answer boxes in your choice.

A. f ( ) (Simplify your answer. Type an exact answer.) ′′ 0 =

B. f ( ) is undefined. ′′ 0

Select the correct choice below and fill in any answer boxes in your choice.

A. f ( ) ′′ 6 ≈ (Type an integer or a decimal. Do not round until the final answer. Then round to two decimal places as needed.)

B. f ( ) is undefined. ′′ 6

Find , the third derivative of f, and , the fourth derivative of f.f (x)′′′ f (x) (4)

f(x) = 4x + 2x + 5x + 2x + 7 4 3 2

Find the third derivative.

f (x)′′′ =

Find the fourth derivative.

f (x)(4) =

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Find the open intervals where the function f(x) is concave upward or concave downward. Find any inflection points.

= − 2x + 6x + 165x − 63 2

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A. The function has a point of inflection at . (Type an ordered pair. Use a comma to separate answers as needed. Use integers or fractions for any numbers in the expression.)

B. The function does not have an inflection point.

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A. The function is concave upward on the interval(s) . (Type your answer in interval notation. Use a comma to separate answers as needed.)

B. The function is never concave upward.

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A. The function is concave downward on the interval(s) . (Type your answer in interval notation. Use a comma to separate answers as needed.)

B. The function is never concave downward.

If a cannonball is shot directly upward with a velocity of ft per second, its height above the ground after t seconds is given by s(t) . Find the velocity and the acceleration after t seconds. What is the maximum height the cannonball reaches? When does it hit the ground?

304 = 304t − 16t2

The velocity after t seconds is v(t) ft/sec.=

The acceleration after t seconds is a(t) .= ft / sec2

The cannonball reaches a maximum height of ft.

The cannonball hits the ground after t sec.=

Find the absolute maximum and minimum values of the following function over the indicated interval, and indicate the x-values at which they occur.

f(x) ; [ , ]= x + x − 12x + 2 1 3

3 1 2

2 − 5 5

The absolute maximum value is at x .= (Use a comma to separate answers as needed. Round to two decimal places as needed.)

The absolute minimum value is at x .= (Use a comma to separate answers as needed. Round to two decimal places as needed.)

Find the absolute extrema of the function f(x) (x ) on the interval [ , ].= 2 + 1 2 / 5

− 3 4

The absolute maximum occurs at x .= (Simplify your answer. Use a comma to separate answers as needed.)

The absolute minimum occurs at x .= (Simplify your answer. Use a comma to separate answers as needed.)

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Find the equation of the tangent line at the given point on the following curve.

, ( , )x + y = 132 2 2 3

The equation of the tangent line to the point ( , ) is y .2 3 =

Find the equation of the tangent line at the given value of x on the curve.

; xy + xy − 25 = x + 2y3 2 2 = 2

y =

Assume x and y are functions of t. Evaluate for the following. dy dt

; , x , yy = 2x + 13 4 dx dt

= 5 = 1 = 2

(Round to two decimal places as needed.) dy dt

=

Assume x and y are functions of t. Evaluate for  

xy x y , with the conditions , x , y . dy dt

3 − 2 + 6 3 = − 36 dx dt

= − 24 = 6 = − 1

dy dt

=

(Type an exact answer in simplified form.)