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MTH 280 Wk 4 – Midterm Exam

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MTH 280 Wk 4 – Midterm Exam

Question 1

A research lab grows a type of bacterium in culture in a circular region. The radius of the circle, measured in centimeters, is given by begin mathsize 12px style r left parenthesis t right parenthesis space equals space 5 space minus space fraction numerator 6 over denominator t squared plus 3 end fraction end style, where t is time measured in hours passed since a circle of a 1cm radius of the bacterium was put into the culture. Express the area, A open parentheses t close parentheses, of the bacteria as a function of time, and find the approximate area of the bacterial culture in 4 hours.

A open parentheses t close parentheses space equals space pi left parenthesis 5 minus fraction numerator 6 over denominator t squared plus 3 end fraction right parenthesis and Area after 4 hours = 14.72 comma space c m squared

A open parentheses t close parentheses space equals space 2 pi left parenthesis 5 minus fraction numerator 6 over denominator t squared plus 3 end fraction right parenthesis and Area after 4 hours = 29.43 comma space c m squared

Question 2

A minivan was purchased for $32,000. If the value of the minivan depreciates by $1,700 per year, find a linear function that models the value V of the car after t years. Use the function and find the value of the car after 5 years.

V(t) = -1,700t + 32,000; value after 5 years = $23,500

V(t) = -1,700 + 32,000; value after 5 years = $30,300

V(t) = 1,700 + 32,000; value after 5 years = $33,700

V(t) = 1,700t + 32,000; value after 5 years = $40,500

Question 3

Simplify tan x left parenthesis c s c x space minus space sin x right parenthesis space.

sin x

cos x

Question 4

A pendulum moving in simple harmonic motion is modelled by the function s open parentheses t close parentheses equals negative 5 cos left parenthesis begin inline style fraction numerator pi t over denominator 4 end fraction end style right parenthesis , where s is measured in inches and t is measured in seconds. Determine the first time when the distance moved is 4 inches.

0.82 s

2.6 s

3.2 s

9.9 s

Question 5

Solve 2 log subscript 5 open parentheses square root of x close parentheses minus log subscript 5 open parentheses 6 x minus 1 close parentheses equals 0.

x equals 1 over 6

x equals 1 fifth

x equals 5

x equals 6

Question 6

Estimate the slope of the tangent line to f open parentheses x close parentheses equals x cubed at x equals 2 by finding the slope of the secant line through left parenthesis 2 comma space f left parenthesis 2 right parenthesis right parenthesis and left parenthesis 2.001 comma space f left parenthesis 2.001 right parenthesis right parenthesis.

0.012

4.002

16.012

Question 7

The following image shows the graph of the function f left parenthesis x right parenthesis equals square root of 4 minus x squared end root:

Find the area between the x-axis and the graph of f left parenthesis x right parenthesis over the interval of open square brackets negative 2 comma space 2 close square brackets using the shaded squares.

6.27 square units

7.5 square units

12.57 square units

8 square units

Question 8

Evaluate limit as x rightwards arrow 5 to the power of minus of space f left parenthesis x right parenthesis if f left parenthesis x right parenthesis space equals space fraction numerator 1 over denominator left parenthesis x minus 5 right parenthesis to the power of 4 end fraction.

negative infinity

negative 10 to the power of negative 4 end exponent

infinity

Question 9

Consider the function f left parenthesis x right parenthesis shown in the following image:

Evaluate limit as x rightwards arrow 0 to the power of minus of space f left parenthesis x right parenthesis.

negative 1

D o e s space n o t space e x i s t

Question 10

Use the Squeeze Theorem to evaluate limit as x rightwards arrow 0 to the power of minus of space f left parenthesis x right parenthesis, where f left parenthesis x right parenthesis equals x squared cos left parenthesis 3 over x right parenthesis.

negative infinity

infinity

Question 11

Evaluate limit as theta rightwards arrow 0 of space f left parenthesis theta right parenthesis, where f left parenthesis theta right parenthesis equals fraction numerator sin theta minus cos 3 theta over denominator theta times left parenthesis 1 plus cos 2 theta right parenthesis end fraction.

1 half

infinity

Question 12

Find the intervals over which the function f left parenthesis x right parenthesis equals fraction numerator 2 x squared plus 3 x plus 1 over denominator x squared minus 5 x end fraction is continuous.

left parenthesis negative infinity comma space 5 right parenthesis space a n d space left parenthesis 5 comma space infinity right parenthesis

left parenthesis negative infinity comma space 0 right parenthesis space a n d space left parenthesis 0 comma space infinity right parenthesis

left parenthesis negative infinity comma space 0 right parenthesis comma space left parenthesis 0 comma space 5 right parenthesis space a n d space left parenthesis 5 comma space infinity right parenthesis

left parenthesis negative infinity comma space minus 5 right parenthesis comma space left parenthesis negative 5 comma space 0 right parenthesis comma space a n d space left parenthesis 0 comma space infinity right parenthesis

Question 13

Find a non-zero value for the constant k that makes the function f left parenthesis x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator tan space k x over denominator x end fraction comma space end cell row cell 5 x plus 3 comma space end cell end table close open table attributes columnalign right end attributes row cell x less than 0 end cell row cell x greater or equal than 0 end cell end table close curly brackets continuous at x equals 0.

1 third

infinity

Question 14

Determine the value of delta greater than 0 for which limit as x rightwards arrow 4 of left parenthesis 3 x squared minus 1 right parenthesis equals 47 space.

delta equals m a x left curly bracket negative 4 minus square root of 16 minus epsilon over 3 end root comma space minus 4 plus square root of 16 plus epsilon over 3 end root right curly bracket

delta equals m i n left curly bracket negative 4 plus square root of 16 minus epsilon over 3 end root comma space minus 4 minus square root of 16 plus epsilon over 3 end root right curly bracket

delta equals m a x left curly bracket 4 minus square root of 16 minus epsilon over 3 end root comma space 4 plus square root of 16 plus epsilon over 3 end root right curly bracket

delta equals m i n left curly bracket 4 minus square root of 16 minus epsilon over 3 end root comma space minus 4 plus square root of 16 plus epsilon over 3 end root right curly bracket

Question 15

Determine the value of delta greater than 0 for which limit as x rightwards arrow 2 of fraction numerator 1 over denominator left parenthesis x minus 2 right parenthesis squared space end fraction equals infinity.

delta equals fraction numerator 1 over denominator square root of M end fraction

delta equals 1 over M

delta equals square root of M

delta equals M

Question 16

A store determines that the daily profit on pens obtained by charging s dollars per pen is P left parenthesis s right parenthesis equals negative 10 s squared plus 75 s minus 5. If the store currently charges 50 cents for a pen, find the rate of change of profit.

$30

$60

$65

$85

Question 17

A ball is dropped from the top of a building that is 15 m high. The position of the ball after t seconds is given by the equation s left parenthesis t right parenthesis equals negative 4.9 t squared plus 15. Find the instantaneous acceleration of the ball after t seconds.

negative 4.9 semicolon space m divided by s squared

negative 9.8 semicolon space m divided by s squared

negative 9.8 t semicolon space m divided by s squared

negative 4.9 t semicolon space m divided by s squared

Question 18

Find fraction numerator d over denominator d x end fraction left parenthesis 3 over x squared plus x left parenthesis x minus 1 right parenthesis right parenthesis.

fraction numerator negative 3 over denominator x squared end fraction plus 2 x minus 1

fraction numerator negative 6 over denominator x cubed end fraction plus 1

fraction numerator negative 3 over denominator x squared end fraction plus 1

fraction numerator negative 6 over denominator x cubed end fraction plus 2 x minus 1

Question 19

The price p (in dollars) and the demand x for an item is given by the price-demand function p left parenthesis x right parenthesis equals 15 minus 0.0015 x. Find the marginal revenue at x equals 1 comma 000.

$1.50

$12.00

$13.50

$15.00

Question 20

Find the equation of the tangent line to f left parenthesis x right parenthesis equals 3 c s c x plus x sin x at x equals straight pi over 2.

y equals x minus 3

y equals x plus 3 plus pi over 2

y equals x plus 3

y equals x plus 3 minus pi over 2

Question 21

A bag of sand hanging from a vertical spring is in simple harmonic motion as given by the position function s left parenthesis t right parenthesis equals negative sin left parenthesis straight pi over 2 t plus straight pi over 6 right parenthesis where t is measured in seconds and s is in inches. Find the velocity of the spring at t equals 2 s.

negative fraction numerator square root of 3 straight pi end root over denominator 4 end fraction space i n c h divided by s

straight pi over 4 i n c h divided by s

fraction numerator square root of 3 straight pi end root over denominator 4 end fraction space i n c h divided by s

straight pi over 2 i n c h divided by s

Question 22

Find fraction numerator d y over denominator d x end fraction if y equals tan to the power of negative 1 end exponent left parenthesis square root of x cubed right parenthesis end root.

fraction numerator 1 over denominator 1 plus x cubed end fraction

fraction numerator 1 over denominator 2 square root of x cubed end root left parenthesis 1 plus x cubed right parenthesis end fraction

fraction numerator 3 x squared over denominator left parenthesis 1 plus x cubed right parenthesis end fraction

fraction numerator 3 square root of x over denominator 2 left parenthesis 1 plus x cubed right parenthesis end fraction

Question 23

The volume of a right circular cylinder of radius x and height y is given by v equals πx squared straight y. Suppose that the volume of the cylinder is constant at 250 πcm cubed. Find fraction numerator d y over denominator d x end fraction when x equals 5 and y equals 10.

negative 100

negative 4

250

Question 24

Find fraction numerator d y over denominator d x end fraction if y equals 5 x left parenthesis cos x right parenthesis to the power of x over 2 end exponent.

5 over 2 left parenthesis cos x right parenthesis to the power of x over 2 end exponent open square brackets 1 over x plus x over 2 c o t x plus fraction numerator ln left parenthesis cos x right parenthesis over denominator 2 end fraction close square brackets

5 x left parenthesis cos x right parenthesis to the power of x over 2 end exponent open square brackets 1 over x minus x over 2 tan x plus fraction numerator ln left parenthesis cos x right parenthesis over denominator 2 end fraction close square brackets

fraction numerator negative 5 x squared sin x over denominator 2 end fraction left parenthesis cos x right parenthesis to the power of x over 2 end exponent minus 1 plus 5 left parenthesis cos x right parenthesis to the power of x over 2 end exponent

fraction numerator negative 5 x squared over denominator 2 end fraction left parenthesis sin x right parenthesis to the power of x over 2 end exponent minus 1 plus 5 left parenthesis cos x right parenthesis to the power of x over 2 end exponent

Question 25

Find the derivative of fraction numerator e to the power of x minus 1 over denominator e to the power of x plus 1 end fraction.

fraction numerator negative 2 e to the power of x over denominator left parenthesis e to the power of x plus 1 right parenthesis squared end fraction

fraction numerator negative e to the power of 2 x end exponent over denominator left parenthesis e to the power of x plus 1 right parenthesis squared end fraction

fraction numerator 2 e to the power of x over denominator left parenthesis e to the power of x plus 1 right parenthesis squared end fraction

fraction numerator e to the power of 2 x end exponent over denominator left parenthesis e to the power of x plus 1 right parenthesis squared end fraction

Question 26

If f left parenthesis x right parenthesis equals square root of 1 plus 8 x end root, find f space apostrophe left parenthesis a right parenthesis for a equals 3 and the equation of the tangent line to f left parenthesis x right parenthesis at x equals a.

f space apostrophe left parenthesis a right parenthesis equals 5 over 4; equation of the tangent line is y equals fraction numerator 5 x minus 13 over denominator 4 end fraction

f space apostrophe left parenthesis a right parenthesis equals negative 4 over 5; equation of the tangent line is y equals fraction numerator negative 4 x plus 13 over denominator 5 end fraction

f space apostrophe left parenthesis a right parenthesis equals 4 over 5; equation of the tangent line is y equals fraction numerator 4 x plus 13 over denominator 5 end fraction

f space apostrophe left parenthesis a right parenthesis equals negative 5 over 4; equation of the tangent line is y equals fraction numerator negative 5 x minus 13 over denominator 4 end fraction

Question 27

The position of a moving particle as a function of time is given by s left parenthesis t right parenthesis equals 1 third t cubed minus t plus 1, where s is in meters and t is in seconds. Find the time at which the particle is at rest and the acceleration of the particle at t equals 3 s.

Time at which the particle is at rest equals 2 space s; acceleration of the particle equals 2 space m divided by s squared

Time at which the particle is at rest equals 1 space s; acceleration of the particle equals 6 m divided by s squared

Time at which the particle is at rest equals 2 space s; acceleration of the particle equals 1 space m divided by s squared

Time at which the particle is at rest equals 3 space s; acceleration of the particle equals 3 space m divided by s squared

Question 28

A car moves on a straight road. The car’s position at time t is given by s left parenthesis t right parenthesis equals t plus sin t, where s is in meters and t is in seconds. Find the acceleration at t equals straight pi over 4 semicolon space s.

negative 1 half semicolon space m divided by s squared

negative fraction numerator 1 over denominator square root of 2 end fraction semicolon space m divided by s squared

fraction numerator 1 over denominator square root of 2 end fraction semicolon space m divided by s squared

1 semicolon space m divided by s squared

Question 29

Find fraction numerator d over denominator d x end fraction open parentheses fraction numerator 1 plus sin squared x over denominator x plus tan cubed 15 x end fraction close parentheses.

fraction numerator 2 sin x cos x over denominator 1 plus 45 tan squared 15 x end fraction

fraction numerator left parenthesis x plus tan cubed 15 x right parenthesis left parenthesis 2 sin x cos x right parenthesis minus left parenthesis 1 plus sin squared x right parenthesis left parenthesis 1 plus 45 tan squared 15 x s e c squared 15 x right parenthesis over denominator left parenthesis x plus tan cubed 15 x right parenthesis squared end fraction

fraction numerator left parenthesis x plus tan cubed 15 x right parenthesis left parenthesis 2 sin x cos x right parenthesis minus left parenthesis 1 plus sin squared x right parenthesis left parenthesis 1 plus 45 tan squared 15 x right parenthesis over denominator left parenthesis x plus tan cubed 15 x right parenthesis squared end fraction

fraction numerator 1 plus 2 sin x cos x over denominator 1 plus 15 tan squared 15 x end fraction

Question 30

Find fraction numerator d f space to the power of negative 1 end exponent over denominator d x end fraction for f left parenthesis x right parenthesis equals fraction numerator x plus 3 over denominator 2 x plus 5 end fraction at x equals 3.

negative 1 over 25

negative 25

1 over 81