Homework 4 PDF

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Fall 2015 Calculus 1 Homework #4 Professor Songjaja...

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Calculus 1 (Math-UA-121)

Fall 2015

Homework 4 Due: Friday, October 2 at the start of class Give complete, well-written solutions to the following exercises. Make sure to write your name and discussion section number, and to staple your solution. 1. Evaluate the following limits. (a)

lim √

x→−∞

4x − 3 25x2 + 4x

(b) lim

x→∞

√

4x4 + 9x − 2x2



2. Sketch the graph of an example of a function f that satisfies all of the following conditions: • f (0) = 3

• lim f (x) = 4

• lim f (x) = −∞

• lim f (x) = ∞ +

• lim− f (x) = −∞

• lim f (x) = 3

x→−∞

x→4

x→0−

• lim+ f (x) = 2 x→0

x→∞

x→4

3. Find the horizontal and vertical asymptotes of the following curve: F (x) = √

x−9 . 4x2 + 3x + 2

4. Let P (n) denote the perimeter of an n-gon inscribed in a circle of radius 1. (See the following figure.) n=3

n=4

n=5

n=6

n=7

n=8

(a) Explain, intuitively, why P (n) approaches 2π as n → ∞.   (b) Show that P (n) = 2n sin nπ .   (Hint: Start by thinking about how to show that P (3) = 6 sin 3π .) π n (c) Combine (a) and (b) to conclude that lim sin =1 n→∞ π n sin θ (d) Use this to give an argument that lim = 1. θ→0 θ 1

Calculus 1 (Math-UA-121)

Fall 2015

5. The graph shows the position function of a car. Use the shape of the graph to explain your answers to the following questions. (a) What was the initial velocity of the car? (b) Was the car going faster at B or C ? (c) Was the car slowing down or speeding up at A, B, and C ? (d) What happened between D and E ?

6. Suppose F (x) =

5x . 1 + x2

(a) Find F ′ (2) and (b) use it to find an equation of the tangent line to the curve y=

5x 1 + x2

at the point (2, 2). 7. Suppose the tangent line to y = f (x) at the point (4, 3) passes through the point (0, 2), find f (4) and f ′ (4). 8. If an arrow is shot upward on the moon with a velocity of 58 m/s, it height (in meters) after t seconds is given by H = 58t − 0.83t2 . (a) Find the velocity of the arrow after one second. (b) Find the velocity of the arrow when t = a. (c) When will the arrow hit the moon? (d) What is the velocity of of the arrow when it hits the moon?

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