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7 Solved Questions on Calculus I for Exam 2 | MATH 111, Exams of Calculus

Material Type: Exam; Class: Calculus I; Subject: Mathematics; University: Colgate University; Term: Fall 1999;

Typology: Exams

Pre 2010

Uploaded on 08/18/2009

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October 18, 1999
Math 111 E and H Exam II
Show all work clearly for partial credit. Do not use the graphing capabilities of your calculator.
1. (36 points) Find dy/dx:
(a) y=xsin x(b) y=x+ 5
x2+ 1 (c) y= tan23x
(d) y=x2
2x(e) x3y2ey= 5 (in terms of xand y) (f) y= (x2+ 2)x
2. (15 points) (a) If y=ln x, what is y0?
(b) Find the equation of the tangent line to y= ln xat x=e4.
3. (15 points) The distance units on a number line are centimeters (cm). The position of a point
on the line at time tsec is given by s(t) = t4
24t2+ 8t+ 5. At what (positive) time(s) is
(are) the acceleration of the point equal to 0, and what are the position and velocity at this
time (these times)? Include units.
4. (10 points) If the edge of a cube is growing at 4 inches/min, how quickly is the total surface
of the cube increasing when the edge is 20 inches long?
5. (10 points) Two of these functions A,B,C,D are the first and second derivatives of another of
these, and the remaining function is unrelated to the others. Which of these is f, which is
f0, which is f”, and which is the unrelated g, and why?
6. (9 points) Use the derivative rules for sin xand cos xto derive the derivative rule for cot x.
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Download 7 Solved Questions on Calculus I for Exam 2 | MATH 111 and more Exams Calculus in PDF only on Docsity!

October 18, 1999 Math 111 E and H — Exam II

Show all work clearly for partial credit. Do not use the graphing capabilities of your calculator.

  1. (36 points) Find dy/dx: (a) y = x sin x (b) y =

x + 5 x^2 + 1

(c) y = tan^2 3 x

(d) y = x^2 − 2 x^ (e) x^3 y − 2 ey^ = 5 (in terms of x and y) (f) y = (x^2 + 2)x

  1. (15 points) (a) If y =

ln x, what is y′? (b) Find the equation of the tangent line to y = ln x at x = e^4.

  1. (15 points) The distance units on a number line are centimeters (cm). The position of a point on the line at time t sec is given by s(t) = t^4 − 24 t^2 + 8t + 5. At what (positive) time(s) is (are) the acceleration of the point equal to 0, and what are the position and velocity at this time (these times)? Include units.
  2. (10 points) If the edge of a cube is growing at 4 inches/min, how quickly is the total surface of the cube increasing when the edge is 20 inches long?
  3. (10 points) Two of these functions A,B,C,D are the first and second derivatives of another of these, and the remaining function is unrelated to the others. Which of these is f , which is f ′, which is f ”, and which is the unrelated g, and why?
  4. (9 points) Use the derivative rules for sin x and cos x to derive the derivative rule for cot x.
  1. (5 points) Suppose y = F (x) is a function for which F ′(x) = l/(x^4 + 1). Then F (x) is one-to-one, so it has an inverse F −l(x). Express (F −^1 )′(x) in terms of F −^1 (x). (Hint: From y = F −^1 (x) we get x = F (y). Differentiate implicitly.)

Some possibly useful equations:

y − y 0 = m(x − x 0 ) A = πr^2 V = e^3

A = 6e^2 A =

bh