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Exam III Problems for Calculus I - Fall 2008 | MATH 111, Exams of Calculus

Material Type: Exam; Professor: Moorhouse; Class: Calculus I; Subject: Mathematics; University: Colgate University; Term: Fall 2008;

Typology: Exams

Pre 2010

Uploaded on 08/17/2009

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Math 111 Calculus 1 Practice Problems for Exam 3 Fall 2008
1. Find the extreme values of f(x) = (x22x)3on the interval [1,3].
(a) local maxima: minima:
(b) absolute maximum: minimum:
2. Evaluate the following limits, where possible. (Hint: Determine whether the limit has
an indeterminate form before doing any calculations.)
Clearly indicate every application of L’Hospital’s Rule!
(a) lim
x0
sin xx
x3
(b) lim
t0(sec xtan x)
(c) lim
x→∞
x2ex
(d) lim
x0x2ln x2
(e) lim
t→∞
tln t
(f) lim
x0
2x1
x
(g) lim
x0
arcsin x
x
(h) lim
x1
(ln x)2
x
3. Let f(x) = 4(x+ 1)
(x+ 2)2.
(a) Find the xand yintercepts of f.
(b) Find the vertical and horizontal asymptotes of f.
(c) Find the intervals on which fis increasing or decreasing.
(d) Find all the points at which fhas a local maximum or a local minimum.
(e) Determine the intervals on which fis concave up or concave down, and find the
inflection points.
(f) Sketch the graph of f. Be sure to label all the xand yvalues of the intercepts,
local maxima, local minima, inflection points, and asymptotes.
4. Let f(x) = (x22x+ 2)ex. Repeat problem3
1
pf2

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Math 111 – Calculus 1 Practice Problems for Exam 3 Fall 2008

  1. Find the extreme values of f (x) = (x^2 − 2 x)^3 on the interval [− 1 , 3]. (a) local maxima: minima: (b) absolute maximum: minimum:
  2. Evaluate the following limits, where possible. (Hint: Determine whether the limit has an indeterminate form before doing any calculations.) Clearly indicate every application of L’Hospital’s Rule!

(a) lim x→ 0 sin^ xx^3 −^ x (b) lim t→ 0 (sec x − tan x) (c) (^) xlim→∞ x^2 e−x (d) lim x→ 0 x^2 ln x^2

(e) (^) tlim→∞ t − ln t

(f) lim x→ 02

x (^) − 1 x (g) lim x→ 0 arcsinx^ x

(h) lim x→ 1 (ln^ x)

2 x

  1. Let f (x) = 4((xx + 2)^ + 1) 2.

(a) Find the x and y intercepts of f. (b) Find the vertical and horizontal asymptotes of f. (c) Find the intervals on which f is increasing or decreasing. (d) Find all the points at which f has a local maximum or a local minimum. (e) Determine the intervals on which f is concave up or concave down, and find the inflection points. (f) Sketch the graph of f. Be sure to label all the x and y values of the intercepts, local maxima, local minima, inflection points, and asymptotes.

  1. Let f (x) = (x^2 − 2 x + 2)ex. Repeat problem

1

  1. (a) Show that f (x) = 2 − x − x^3 has exactly one root. (b) Let f (x) be a function that is differentiable for all x. Suppose that f (0) = −3, and f ′(x) ≤ 5 for all values of x. How large can f (2) possibly be? (You must use the Mean Value justify your answer, even if you can do it in your head.)
  2. Suppose that g(x) is a continuous function on [0, 9], whose derivative is shown (see page 305 of Stewart). Answer the following questions, explaining your answers fully. (a) For what values of x is g(x) increasing? For what values of x is g(x) decreasing? (b) For what values of x (if any) does g(x) have a vertical tangent or a vertical cusp? (c) For what values of x is g(x) concave up? For what values of x is g(x) concave down? (d) Sketch a graph of f assuming that f (0) = 0.
  3. (a) Find the dimensions of the largest rectangle that can be inscribed in an equilateral triangle of side length 4 units. (b) A farmer wants to build a rectangular pen and then split it into 3 pens of equal size as shown below. If the farmer has 800 feet of fencing, what is the maximum possible total area that can be enclosed? What are the dimensions of each pen in that case? (c) A cylindrical can is made to contain 2000cm^3 of liquid. Find the dimensions of the can that will minimize the cost of making the can, by minimizing the amount of material needed. (d) A steel pipe is carried down a hallway 9 ft wide. At the end of the hallway is a right turn into a narrower hallway 6 ft wide. What is the length of the longest pipe that can be carried horizontally around the corner (see the picture on page 332 #66.)
  4. (a) Find the most general antiderivative of the function f (x) = x^3 + sin x + sec^2 x + 3 ex^ + (^) 1+^1 x 2 (b) Find the function f if f ′(x) = √ 13 −x 2 + x, and f (0) = 2 (c) Find the function f if f ′′(x) = 3et^ + sin t, f (0) = 1, f (π) = 0
  5. A car is traveling at 100km/h when the driver applies the brakes. The car decelerates at a constant rate of 20 km/s^2. What is the distance traveled before the car comes to a complete stop?