# Math 226 Worksheet 12 1 The graph of the derivative of a function by chenboying

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```									                                   Math 226 Worksheet 12
1. The graph of the derivative of a function f (x) is given below.

f’

1        2          3        4

(a) What are the critical points of f (x)?
(b) Make a rough sketch of y = f (x). Be sure to get right the intervals where the function is
increasing and decreasing, and where it levels oﬀ.
(c) Which critical point(s) correspond to relative extrema?
2. Draw a graph of a function f that is concave down on [0, 5].
(a) What can you conclude about the derivative of f ?
(b) What can you conclude (from 2a) about the derivative of f ?
(c) Make an analogous argument about functions whose graphs are concave up on an interval.
3. Weeble Knieval is a stuntman.
(a) Let his height above the ground at time t seconds be h(t). Give meaningful interpretations
to h (t) and h (t).
(b) A simple model of gravity is that it accelerates an object towards the ground at (about) 9.8
meters per second per second, or 9.8 m/s2 . Convince yourself (and me) the units make sense.
(c) Suppose Weeble is only experiencing acceleration due to gravity. Write an formula for h .
What is h (1) − h (0)? Get the units and sign right.
(d) Using your knowledge of h , guess a formula for h .
(e) Your guessed formula in (3d) is not the only possible guess, because if we add any constant
to it, it will not change the derivative. Now let me give you more information. Suppose
Weeble’s vertical velocity at time t = 0 is 4.9 m/s. Find the unique formula for h that ﬁts
the known information.
(f) Using your knowledge of h , guess a formula for h.
(g) Suppose Weeble began on the ground. How high is Weeble after 1 second? Where is he at
time t?
(h) Write and prove a general formula for the height of a “projectile” that experiences vertical
acceleration only due to gravity. Your formula should be in terms of the initial height (at
time t = 0) and initial velocity.
4. Suppose that g and h are increasing functions on an interval I . For the following functions, either
show that they must be increasing on I or give a counter-example.
a) g + h         b) g · h         c) g ◦ h

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