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Indicated Limit - Mathematics - Exam, Exams of Mathematics

Punjab Engineering CollegeMathematics

This is the Exam of Mathematics which includes Price Per Unit, Demand, Commodity, Equation, Marginal Revenue, Total Revenue, Demand, Function, Marginal Revenue etc. Key important points are: Indicated Limit, Certain Logistic Model, Population, Model is Correct, Happen, Approximately, Limit, Definition, Derivative, Slope Intercept

Typology: Exams

2012/2013

Uploaded on 02/25/2013

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MATH 22 FINAL EXAMINATION
Spring Semester 2001 May 4, 2001
NAME: _________________________________________ SCORE: ____________________
INSTRUCTOR: __________________________________ SECTION NUMBER: _________
• Answers are to be given in exact form (e.g., ln(3) and not 1.099,
2
and not 1.41,
4
π
and not
0.79, 1/3 and not 0.333333, etc.) unless decimal approximations are requested. A terminating
decimal number may be written in either decimal or fractional form. For example,
20
29
= 1.45;
either form is acceptable.
• With the exception of symbolic manipulators such as the TI 89-95, calculators (graphing or
scientific) are permitted. The use of stored formulas is prohibited.
• You must use calculus when instructed to do so and for all maximization and minimization
problems. No credit will be given for solutions found by trial and error.
• ALL work is to be shown on this exam paper. No credit will be given for correct answers
without adequate supporting work.
• Place all answers in the spaces provided.
• There are 12 pages (including this one) and 25 problems on the exam. Each question is worth 3
points. Verify now that no problems are missing. If your exam is incomplete, immediately
obtain a complete copy from your instructor. If you turn in an incomplete exam, the missing
problems will receive grades of zero.
• Abide by the Honor Code and, having done so, sign the honor pledge when you finish the exam.
I pledge that I have neither given nor received any unauthorized assistance on this exam.
___________________________________________________________
(signature)
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MATH 22 FINAL EXAMINATION

Spring Semester 2001 May 4, 2001

NAME: _________________________________________ SCORE: ____________________

INSTRUCTOR: __________________________________ SECTION NUMBER: _________

  • Answers are to be given in exact form (e.g., ln(3) and not 1.099, 2 and not 1.41, 4

and not

0.79, 1/3 and not 0.333333, etc.) unless decimal approximations are requested. A terminating decimal number may be written in either decimal or fractional form. For example, 2029 = 1.45; either form is acceptable.

  • With the exception of symbolic manipulators such as the TI 89-95, calculators (graphing or scientific) are permitted. The use of stored formulas is prohibited.
  • You must use calculus when instructed to do so and for all maximization and minimization problems. No credit will be given for solutions found by trial and error.
  • ALL work is to be shown on this exam paper. No credit will be given for correct answers without adequate supporting work.
  • Place all answers in the spaces provided.
  • There are 12 pages (including this one) and 25 problems on the exam. Each question is worth 3 points. Verify now that no problems are missing. If your exam is incomplete, immediately obtain a complete copy from your instructor. If you turn in an incomplete exam, the missing problems will receive grades of zero.
  • Abide by the Honor Code and, having done so, sign the honor pledge when you finish the exam.

I pledge that I have neither given nor received any unauthorized assistance on this exam.

(signature)

  1. Find the indicated limit, if it exists.

lim t^3 − 4t. t → − 2 5t^2 + 11t + 2

  1. According to a certain logistic model, the world’s population (in billions) t years after 1960 will be

approximately (^) t e

P t 0. 08 1 12

=.^ If this model is correct, what will happen to P(t) in the long run

(that is, what is the limit of P(t) as t →∞)?

  1. Find h″(1) if h (x) = x − 4. 3 − 2x
  2. Find the slope-intercept form of the equation of the tangent line to the curve y = f(x) = (25 − 3x 2 ) 3 at the point with x-coordinate 3.

y =

  1. y is defined implicitly as a function of x by the equation x 3 − xy 3 = y − 8. Find (^) dxdy^.
  2. Find h′(x) if h(x) = (50 + x)e – 3x^. Factor your answer completely.
  3. Find g′(x) if g (x) = ln(– 30x + 5x 3 + 14).
  1. The first and second derivatives of a function f are f ′(^ x )= e^2 −^ x^ ( 3 x + 1 )and f ′′^ ( x )= e^2 −^ x^ ( 2 − 3 x ).

List the open interval(s) on which f is both increasing and concave down.

  1. A function f has the following properties: (i) The domain of f is the set of real numbers x for which − 2 ≤ x ≤ 5. (ii) f ′ (x) > 0 for − 2 < x < 0. (iii) f ′ (x) < 0 for 0 < x < 5. (iv) f ′′ (x) < 0 for 3 < x < 5 and (3, 6) is an inflection point.

Which of the graphs shown can be the graph of y = f(x)?

GRAPHS WERE HAND-DRAWN SO YOU DO NOT SEE THEM HERE.

  1. Find values of a and b for which 6

bx

ax y f x has the graph shown for x in (−3, ∞).

a = b =

GRAPH WAS HAND-DRAWN SO YOU DO NOT SEE IT HERE.

  1. A manufacturer’s total monthly revenue is R(q) = 240q + 0.05q^2 dollars, where q units are produced during the month. Currently the manufacturer is producing 80 units a month and is planning to decrease the monthly output by 0.65 unit. Use calculus (specifically, the method of approximation by differentials) to estimate the resulting change in total monthly revenue.

≅R ≈ dollars

  1. Use calculus to find the critical numbers of f(x) = (x + 2 )2/3^ in the interval [− 10, −1]. Find the absolute minimum and maximum values of f(x) on that interval.

minimum:

maximum:

edition tend to decrease exponentially. At the time publicity was discontinued, I Survived “Survivor” was experiencing sales of 25,000 copies per month. Two months later, sales of the book had dropped to 4,000 copies per month. What will the sales be after 3 more months? Answer to the nearest 50 copies.

copies

  1. Find dt t t

∫ t 

  1. Find dx x

x

4

3 .

  1. Evaluate dx

∫ x

−

−

1

3

2

5 using the Fundamental Theorem of Calculus.

  1. Find the general solution of y e x dx

dy (^) 2 2 = 4 −.

  1. Use calculus to find the area of the region bounded by the curve y = 25 – x^2 and the x-axis. Include a rough sketch of the region.