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Differentiate - Calculus One - Exam, Exams of Calculus

Gandhi Institute of Technology and ManagementCalculus

This is the Exam of Calculus One which includes Domain, Limits, Function, Absolute Extrema, Area, Shaded Region, Average Value, Function, Equation, Line Tangent etc. Key important points are: Differentiate, Logarithmic Differentiation, Equation, Satisfies, Absolute Minimum Value, Area, Shaded Region, Bounded, Right, Line

Typology: Exams

2012/2013

Uploaded on 02/25/2013

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APPM 1345 Exam 3 Spring 2012
INSTRUCTIONS: Books, notes, and electronic devices are not permitted. Write (1) your name and
(2) a grading table on the front of your bluebook. Start each problem on a new page. Simplify
your answers. A correct answer with incorrect or no supporting work may receive no credit, while
an incorrect answer with relevant work may receive partial credit. Unless otherwise indicated, show
all work.
1. (18 points) Differentiate the following functions.
(a) y= ln(ln 5x))
(b) y= (ln x)x
(c) y= arctan(e2x)
2. (18 points) Evaluate the following integrals.
(a) Z1
xlog8xdx
(b) Zx3(x2)dx
(c) Zln(π/2)
ln(π/6)
2excos(ex)dx
3. (12 points) Evaluate the following expressions.
(a) cot(cos−1√3
2) =
(b) sec(sin−1x) =
(c) lim
x→−∞ tan−1(ex) =
4. (8 points) Let y=2√x3+ 3
x3
√x+ 7 . Use logarithmic differentiation to find y0. Leave your answer
unsimplified.
5. (8 points) Find the values of λfor which y=eλx satisfies the equation y00 = 6y0−9y.
6. (12 points) Let f(x) = xln 2x−xfor x > 0.
(a) Find f0and f00.
(b) Find the absolute minimum value of f.
pf2

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APPM 1345 Exam 3 Spring 2012

INSTRUCTIONS: Books, notes, and electronic devices are not permitted. Write (1) your name and (2) a grading table on the front of your bluebook. Start each problem on a new page. Simplify your answers. A correct answer with incorrect or no supporting work may receive no credit, while an incorrect answer with relevant work may receive partial credit. Unless otherwise indicated, show all work.

  1. (18 points) Differentiate the following functions.

(a) y = ln(ln 5x)) (b) y = (ln x)x (c) y = arctan(e^2 x)

  1. (18 points) Evaluate the following integrals.

(a)

x log 8 x

dx

(b)

x 3 (x

(^2) ) dx

(c)

∫ (^) ln(π/2)

ln(π/6)

2 ex^ cos(ex) dx

  1. (12 points) Evaluate the following expressions.

(a) cot(cos−^1

√ 3 2 ) = (b) sec(sin−^1 x) = (c) lim x→−∞ tan−^1 (ex) =

  1. (8 points) Let y =

x^3 + 3 x 3

x + 7

. Use logarithmic differentiation to find y′. Leave your answer unsimplified.

  1. (8 points) Find the values of λ for which y = eλx^ satisfies the equation y′′^ = 6y′^ − 9 y.
  2. (12 points) Let f (x) = x ln 2x − x for x > 0.

(a) Find f ′^ and f ′′. (b) Find the absolute minimum value of f.

  1. (10 points) Find the area of the shaded region shown below, bounded above by the curve y = ex/^2 , below by y = e−x/^2 , and on the right by the line x = 2 ln 2.

y á ex ê^2

y á e - x ê^2

2 lnH 2 L x

y

  1. (14 points) Jack’s beanstalk is growing at a rate proportional to its height. In just four seconds, it grows from 3 ft to 5 ft tall.

(a) How tall will the beanstalk be after 3 more seconds? (b) How long will it take the beanstalk to grow from 3 ft to 300 ft tall? Use the approximations ln 2 ≈ 0. 7 , ln 3 ≈ 1. 1 , and ln 5 ≈ 1. 6 to calculate your answer.