## MTH 263 NVCC Properties of Logarithmic Function Calculus Exam Summary is pretty clear. 10 calculus 1 problems that I need solved with ALL the work shown pl

MTH 263 NVCC Properties of Logarithmic Function Calculus Exam Summary is pretty clear. 10 calculus 1 problems that I need solved with ALL the work shown please. Write on this Test
8/18 – Version A
REV. 1/20
Northern Virginia Community College – NOVA Online
MTH 263 – Calculus I
(Multiple Instructors)
Exam 2_A – Form 8-18-A
(Cover Sheet plus 7 pages)
Instructor: __________________________________________________
Name: _____________________________________________________
EMPLID/SIS #: ____________________________
Date: _____________Testing Location: __________________________
*2 HR Time Limit: Start Time:________ Projected Finish Time:________ Actual Finish Time:________
PROCTOR DIRECTIONS:
Testing Center Provide
Student May Use Own
Student May Bring
and Use:
Paper – Lined or unlined
Calculator – Basic 4function, if available
Calculator – Handheld,
scientific, non-graphing
calculator without
internet access ONLY
document the times
indicated above.
Computer, internet, and
mobile device
calculators are NOT
permitted.
One 8 ½ X 11 inch
sheet of paper with
formulas, theorems
and definitions on
one or both sides.
This is to be
collected and
forwarded to NOVA
Online with the
exam.
Forward to
NOVA Online
Any notes
student brought
and used.
Any and all
papers written
on while testing,
including this
exam.
student write
name on every
page.
Student Directions: Be sure your full name, EMPLID #, MTH 263, and Exam 2A are
on each sheet of paper. When finished, return this exam, the formula sheet, and all
paper provided to the testing center proctor so they may be forwarded to your
Cover Sheet
MTH 263 Form 8-18-A
Exam 2_A
Name:______________________________
Directions: Read each question carefully. Show All of your work to earn full credit. All steps
must be shown! Please make sure your work is neatly written and organized.
1. Find the derivative of the following functions:
(a)
(b)
f ( x) = sin 3 (cos(2 x))
y = ecosh5x
Use the properties of logarithmic function to rewrite y before you find the derivative:
(c)
 x2 + 3 
y = ln 
5
 ( 2 x + 5 ) 
(d)
Find f (4) ( x) for f ( x) =
1
x
1
2. Use the given graph of the derivative f ′ of a continuous function f over the interval
( 0,9 ) .
To find the following:
(a) On what interval(s) is f increasing?
(b) On what interval(s) is f decreasing?
(c) At what values of x does f have a local maximum or minimum?
3. Use implicit differentiation to find an equation of the tangent line to the graph of
x 2 + 4 xy + y 2 =
13 at the point ( 2,1) .
2
4. Let f ( x)= sin( x + π ) + cos 2 x
(a) Compute f ′( x), f ′′( x), f (3) ( x), f (4) ( x)
(b) Observe the pattern in part (a) and use it to compute f (9) (0)
5. (a) A plane flying horizontally at an altitude of 3 mi and a speed of 480 mi/h passes
directly over a radar station. Find the rate at which the distance from the plane
to the station is increasing when it is 4 mi away from the station.
(b) A particle moves with position function s (t ) =t 4 − 4t 3 − 20t 2 + 20t , t ≥ 0 .
At what time does the particle have a velocity of 20 m/s ?
3
6. (a)
Find an equation of the tangent line to the curve at the given point.
y=
4
, (0, 2)
1 + e− x
(b) True or false. If it is false, explain why it is false.
d 2
x + x = 2x +1
dx
=
f ( x) tan
=
x and a
(a) Given
π
4
i. Find the equation of the line that represents the linear approximation
to the function f at the given value a =
π
.
4
ii. Use the linear approximation of f you found in (i) to estimate the function value
π 
f  .
5
4
(b) Find the differential dy given y = sec 2 x
8.
Given the function f ( x) =
(a) all critical numbers
x4 + 1
. Find each of the following.
x2
(b) intervals that f is concave up
(c) classify all local extrema
(d) Sketch the graph of f .
5
9. (a) At 2:00 PM a car’s speedometer reads 30 mi/h. At 2:15 PM it reads 50 mi/h.
Assume the velocity function of the car is differentiable. Use the Mean Value Theorem
to show that at some time between 2:00 and 2:15 the acceleration is exactly
80 mi/h2.
(b) Sketch the graph of a function that satisfies all of the given conditions.
f ′( x) > 0 if x ≠ 2, f ′′( x) > 0 if x < 2, f ′′( x) < 0 if x > 2, f has inflection point ( 2,5 ) ,
=
lim f ( x) 8,=
lim f ( x) 0
x →∞
x →−∞
6
10.
The cost in dollars to manufacture x toys is given by
C ( x) =+
84 0.16 x − 0.0006 x 2 + 0.000003 x3
(a)
Find the marginal cost function C ′( x)
(c) Find C ′(100) and explain its meaning. What does it predict?
(c) Compare C ′(100) with the actual cost of producing the 101st toy.
7