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Math 105: Exam II Preparation - Review of Derivatives, Limits, and Differential Equations, Exams of Calculus

University of MumbaiCalculus

Review questions for exam ii of math 105, covering topics such as finding derivatives, limits, and equations of tangent lines. It also includes problems on differential equations and optimization. Students are encouraged to solve problems (a) to (d) in parts 1 and 2, and all parts in problems 7 and 8.

Typology: Exams

2012/2013

Uploaded on 03/06/2013

ramesh
ramesh 🇮🇳

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Math 105: Review for Exam II
1. Find dy/dx for each of the following.
(a) y=x2+2
x+e2+e2x+ ln 2+ ln (2x) + arctan 2
(b) y=√x·arctan (5x)
(c) y= ln(tan(2cos(x2)))
(d) y= sin5x+eπ
ln 4 + arcsin 6x
2. Consider the curve defined by x3+y3=9
2xy (known as the Folium of Descartes).
(a) Find dy/dx.
(b) Find the equation of the tangent line at the point (1,2).
pf3
pf4
pf5

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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 = sin^5

x + eπ ln 4 + arcsin 6x

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

xy (known as the Folium of Descartes).

(a) Find dy/dx.

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

  1. Evaluate the following limits.

(a) lim x→ 0

sin 3x 5 x

(b) lim x→∞

ex ln x

(c) lim x→ 0

1 − cos 2x 3 x

(d) lim x→ 0 + x^2 ln x [The 8:00 and 9:30 sections may omit this part.]

(e) lim x→ 0

1 − cos 4x 5 x^2

  1. Suppose the domain of f(x) is all reals and that f has an inverse function f−^1 (x). Further, suppose that f(2) = 5 and f′^ (2) = e. Finally, let h(x) = 1/f(x).

(a) What point must be on the graph of f−^1 (x)?

(b) What point must be on the graph of h(x)?

(c) Give an example of a point that cannot be on the graph of f(x). Do not choose a point with x-value of 2.

(d) What is the value of the derivative of h(x) at x = 2?

  1. Suppose that y = f(t) is a solution to the differential equation y′^ =

π arcsin t+y^2 and that f

. Find the equation of the tangent line to f at

  1. How would your answers to the previous question change if the domain of f were [− 10 , 10]?
  2. Use Newton’s Method with an initial guess of x 0 = −1 to find the next three approximations to a solution of x^3 + x + 1 = 0. Then test your final approximation to see if it appears to be close to a root.
  3. Circle always, sometimes, or never to make each statement below correct.

(a) If f′(1) = 0 then f always/sometimes/never has a critical point at x = 1.

(b) If f′(2) = 0 then f always/sometimes/never has a local maximum or local minimum at x = 2.

(c) If x = 3 is a critical point of f, then f′^ (3) is always/sometimes/never 0.

(d) If f′′(4) = 0, then f always/sometimes/never has an inflection point at x = 4.

(e) If f has a global maximum at x = 5, then f′^ (5) is always/sometimes/never 0.

(f) If f′(6) = 0 and f′′(6) = −2, then f always/sometimes/never has a local maximum at x = 6.

(g) If f′(7) = 0 and f′′(7) = 0, then f always/sometimes/never has a local extremum at x = 7.

  1. The rate of change of a population P (t) of eels is proportional to the size of the population. When the population is 40000, it is growing at a rate of 400 eels per year. At time t = 0, the population is 10000.

(a) Write a differential equation whose solution is P (t).

(b) Solve your differential equation.

(c) When will the population reach 60000?

  1. You are designing an 18 ft^3 box that will have a square bottom and no top. The material for the bottom costs 40 cents per square foot and the material for the sides costs 30 cents per square foot. What dimensions give the least total cost? Be sure to show how you know you have found the minimum.