Math F���: Midterm � Review November ��, ����

�. Consider the function f (x) and its derivatives:

f (x) = ex� + x

f ′(x) = xex� + x�

f ′′(x) = ex(x� + �)(� + x)� .

a. Find the critical numbers of f (x).

b. Find the open intervals on which the function is increasing or decreasing.

c. Find the open intervals on which the function is concave up or concave down.

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Math F���: Midterm � Review November ��, ����

d. Classify all critical points – using the �rst derivative test.

e. Classify all critical points – using the second derivative test.

f . Find the in�ection points.

g. Sketch the graph.

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Math F���: Midterm � Review November ��, ����

�. Find the linearization of f (x) =√x at a = � and use it to estimate√

�.�.

�. Show that the point (�, �) lies on the curve x� + xy − y� = �. �en �nd the slope of thetangent line to the curve at that point.

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Math F���: Midterm � Review November ��, ����

�. A ball of metal is being heated in an oven, and its radius is increasing at a rate of �.�cm/min. At what rate is the ball’s volume increasing when its radius is � cm?

�. Evaluate the following limits.

limx→�

� + x − exsin x limx→�+(� + �x)��x

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Math F���: Midterm � Review November ��, ����

�. A landscape architect wishes to enclose a rectangular garden on one side by a brick wallcosting ��� per foot and on the other three sides with a metal fence costing ��� per foot.�e area of the garden is to be �����. What are the dimensions of the garden that mini-mize the cost of the fencing?

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Cost : C = Boyt

20×+104=40-11-20

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Area : A = xy= 800

y = 801CEO : 160¥ =/

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Math F���: Midterm � Review November ��, ����

�.

a. State the Mean Value �eorem and draw a picture to illustrate it.

b. Suppose f (x) is continuous on [−�, �] and has a derivative at each x in (−�, �). Iff (−�) = � and f (�) = �, what does the Mean Value �eorem let you conclude?

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or ( a. b ),

then thee is a in ( a,b) where

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