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Math 105: Exam II Preparation - Derivatives, Limits, and Function Analysis, Exams of Calculus

Aliah UniversityCalculus

Review questions for exam ii in math 105, covering topics such as finding derivatives, evaluating limits, and analyzing functions. Questions include finding derivatives of various functions, determining if points lie on a curve, finding equations of tangent lines, evaluating limits, and finding antiderivatives. Additionally, there is a problem involving optimizing the dimensions of a box-shaped aquarium.

Typology: Exams

2012/2013

Uploaded on 03/06/2013

jojy
jojy 🇮🇳

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Math 105: Review for Exam II
1. Find dy/dx for each of the following.
(a) y=x2+ 2x+e2+e2x+ ln 2 + l n (2x) + arctan 2
(b) y=√x·arctan (5x)
(c) y= ln(tan(2cos(x2)))
(d) y=x+eπ
cos 4 + sin5(6x)
2. Consider the curve defined by x3+y3=9
2xy (known as the Folium of Descartes).
(a) Find dy/dx.
(b) Verify that the point (1,2) is on the curve above.
(c) Find the equation of the tangent line at the point (1,2).
pf3
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Math 105: Review for Exam II

  1. Find dy/dx for each of the following.

(a) y = x^2 + 2x^ + e^2 + e^2 x^ + ln 2 + ln (2x) + arctan 2

(b) y =

x · arctan (5x)

(c) y = ln(tan(2cos(x

(^2) ) ))

(d) y =

x + eπ cos 4 + sin^5 (6x)

  1. Consider the curve defined by x^3 + y^3 =

xy (known as the Folium of Descartes).

(a) Find dy/dx.

(b) Verify that the point (1,2) is on the curve above.

(c) Find the equation of the tangent line at the point (1,2).

  1. Evaluate the following limits.

(a) lim x→ 1

x^3 − 1 7 − 7 x

(b) lim x→ 0

1 − cos 2x 3 x

(c) lim x→ 0

1 − cos 4x 5 x^2

(d) lim x→∞

x^2 2 x

  1. Find the following.

(a) an antiderivative of y =

1 − 9 x^2

  • x^3 + cos(2x) + e^3

(b) tan(arccos x) (rewritten as an algebraic expression - no trigonometric functions)

  1. How would your answers to the previous question change if the domain of f were [− 10 , 10]?
  2. You are planning to build a box-shaped aquarium with no top and with two square ends. Your budget is $288. If the glass for the sides costs $12 per square foot and the opaque material for the bottom costs $3 per square foot, what dimensions will maximize the volume? Be sure to show how you know you have found the maximum.