## Math 1300: Calculus I, Instructor: Dana Ernst

### Review for Midterm Exam  #2

Here is a review for your upcoming test.  The test covers sections 3.4 through 4.6.  I will not collect this.  Do what you want with it. This review will give you a good indication of what you will be expected to know for the test.

Here is a list of things that you must know and understand:

• Derivatives of trig functions
• Derivatives of exponential functions with base
• Derivatives of natural logs
• Find horizontal and vertical tangents
• Find equation of tangent line to a graph at a given point
• Find critical points
• Find absolute max and min of a continuous function on a closed interval
• Solve applied max and min problems
• Implicit differentiation
• Related Rates
• Linear approximation
• Differentials
• How to find intervals where a given function is increasing or decreasing
• Rolle’s Theorem
• Mean Value Theorem
• How to find intervals where a given function is concave up or concave down
• First Derivative Test
• How to find inflection points
• Second Derivative Test
• Sketch graphs

Reminders:  During the exam, SHOW ALL YOUR WORK and JUSTIFY YOUR ANSWERS when necessary.  Also, be organized and neat!

Answer each of the following questions.

1. Provide an example of a function f that is continuous at
, but
f is not differentiable at
.

1. Provide an example of a function f such that
, but
f does not have a local maximum or a local minimum at
.

1. Provide an example of a function f such that
, but
f does not have an inflection point at
.

1. Differentiate each of the following functions, but do not simplify.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

1. Find an equation of the tangent line to the graph of
when
.

1. Find an equation of the tangent line to the graph of
when
.

1. Find all points on the graph of
where the tangent line is horizontal or vertical.

1. Determine
for the following implicit equations.

(a)

(b)

1. Find the absolute (global) max and min for
on the interval
.

1. Find the absolute (global) max and min for
on the interval
.
2. Find the point on the graph of
that is closest to the point
.

1. Exercise #11, page 159.

1. Exercise #18, page 160.

1. Assume that a continuous function f has exactly one critical number:
.  In each case below, decide whether the point at which
is a local max, local min, or neither.

(a)    and

(b)

(c)    and

1. Determine if Rolle’s Theorem applies to each of the following functions.  Explain your answer.

(a)

(b)

16.  Given , show that f satisfies the hypotheses of the Mean Value Theorem on the interval , and then find all c in the interval  that satisfy the conclusion of the Mean Value Theorem.

17.  A point moves along the curve  in such a way that the y-value is decreasing at a rate of 2 meters per second.  At what rate is x changing when ?

18.  An interstellar nugget is circling the planet Earth a mile from its surface in a clockwise direction.  An observer is standing at a fixed location watching the nugget with his nuggetscope.  If the nugget is traveling at a rate of 200 mph, then what is the rate of change in the angle of elevation of the nuggetscope when the angle is ?  For this problem, assume the surface of the Earth is “locally” flat (not round).  For this problem, assume that the observer is standing facing the nugget with the nugget approaching.

19.  Determine the intervals where the following functions are increasing, decreasing, concave up, and concave down.

(a)

(b)

20.  Find the coordinates of the local maximum, local minimum, and inflection points of the following functions.

(a)

(b)   on

21.  Sketch the graph of  by finding x-intercepts, y-intercept, intervals of increase and decrease, coordinates of turning points (local max/min), intervals of concavity, coordinates of inflection points.

22.  The position function for a nugget is given by  where t is measured in seconds and  is measured in feet.  Find the acceleration of the nugget when

23.  Find the linear approximation for  at .

24.  Approximate the value of  using a linear approximation.

25.  Find  for each of the following.

(a)

(b)