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

*Proctor – please

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.

Please have

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

instructor for grading.

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

7. Answer the following:

=

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

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