Help **Transcription of Educational Material**---**1. Problem on Planet Mathemagical:**A rock is thrown upward on the planet Mathemagicald with a velocity of 12 m/s. Its height in meters $ t $ seconds later is given by the equation:\[ y = 12t - 1.86t^2 \]a. **Find the average velocity over the given time intervals.** i. $[1, 2]$ ii. $[1, 1.5]$ iii. $[1, 1.01]$ iv. $[1, 1.001]$b. **Estimate the instantaneous velocity when $ t = 1 $.**---**2. Parabola and Perpendicular Bisectors:**The figure below shows a point $ P $ on the parabola $ y = -x^2 $ and the point $ Q $ where the perpendicular bisector of $ QP $ intersects the y-axis. As $ P $ approaches the origin along the parabola, what happens to $ Q $? Does it have a limiting position? If so, find it.**Explanation of the Diagram:**- A graph displays a parabola following the equation $ y = -x^2 $.- Point $ P $ is located on this parabola.- A line (perpendicular bisector of $ QP $) intersects the y-axis, determining point $ Q $.The problem explores the limiting behavior of point $ Q $ as point $ P $ moves toward the origin on the parabola.--- **Problem 3: Graphing a Function**Sketch a graph of a function $ y = f(x) $ that satisfies all of the following conditions (if possible):a. $ \lim_{x \to 0^-} f(x) = 1 $b. $ \lim_{x \to 0^+} f(x) = -1 $c. $ \lim_{x \to 3} f(x) = 0 $d. $ f'(3) $ does not existe. $ f'(-1) = 2 $**Graph Description:**The graph should demonstrate a function with a jump discontinuity at $ x = 0 $, where the left-hand limit (as $ x \to 0^- $) approaches 1 and the right-hand limit (as $ x \to 0^+ $) approaches -1. As $ x \to 3 $, the function should approach 0, but there should be a point or a cusp to indicate that the derivative at $ x = 3 $ does not exist. At $ x = -1 $, the slope of the function, represented by the derivative, should equal 2, indicating a linear tangent.**Problem 4: Function Limit Exploration**Is it possible that there exists some function $ g(x) $ and some constant $ L $ such that\[\lim_{x \to 2^-} g(x) = \lim_{x \to 2^+} g(x) = L\]but \[\lim_{x \to 2} g(x) \]does not exist? If so, draw a sketch of such a function. If no, explain why not.**Explanation:**A function $ g(x) $ can have the left-hand and right-hand limits equal to a constant $ L $, yet the general limit $ \lim_{x \to 2} g(x) $ might not exist if there is a point of discontinuity (such as a hole or a jump at $ x=2 $). The graph should have left-hand and right-hand limits meeting at $ L $, but a disruption at $ x=2 $.

Name: Help **Transcription of Educational Material**---**1. Problem on Plane
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Description: Help **Transcription of Educational Material**---**1. Problem on Planet Mathemagical:**A rock is thrown upward on the planet Mathemagicald with a velocity of 12 m/s. Its height in meters $ t $ seconds later is given by the equation:\[ y = 12t - 1.8

Given: velocity=12m/sy=12t-1.86t2

Math

Calculus

If a rock is thrown upward on the planet Mathemagicland with a velocity of 12 m/s, its height in meters t seconds later is given by y = 12t- 1.86t2. a. Find the average velocity over the given time intervals. [1, 2] [1, 1.5] [1, 1.01] [1, 1.001] i. ii. ii. iv. b. Estimate the instantaneous velocity when t 1.

If a rock is thrown upward on the planet Mathemagicland with a velocity of 12 m/s, its height in meters t seconds later is given by y = 12t- 1.86t2. a. Find the average velocity over the given time intervals. [1, 2] [1, 1.5] [1, 1.01] [1, 1.001] i. ii. ii. iv. b. Estimate the instantaneous velocity when t 1.

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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**Transcription of Educational Material**

---

**1. Problem on Planet Mathemagical:**

A rock is thrown upward on the planet Mathemagicald with a velocity of 12 m/s. Its height in meters $ t $ seconds later is given by the equation:

\[ y = 12t - 1.86t^2 \]

a. **Find the average velocity over the given time intervals.**

i. $[1, 2]$

ii. $[1, 1.5]$

iii. $[1, 1.01]$

iv. $[1, 1.001]$

b. **Estimate the instantaneous velocity when $ t = 1 $.**

---

**2. Parabola and Perpendicular Bisectors:**

The figure below shows a point $ P $ on the parabola $ y = -x^2 $ and the point $ Q $ where the perpendicular bisector of $ QP $ intersects the y-axis. As $ P $ approaches the origin along the parabola, what happens to $ Q $? Does it have a limiting position? If so, find it.

**Explanation of the Diagram:**

- A graph displays a parabola following the equation $ y = -x^2 $.
- Point $ P $ is located on this parabola.
- A line (perpendicular bisector of $ QP $) intersects the y-axis, determining point $ Q $.

The problem explores the limiting behavior of point $ Q $ as point $ P $ moves toward the origin on the parabola.

---

**Problem 3: Graphing a Function**

Sketch a graph of a function $ y = f(x) $ that satisfies all of the following conditions (if possible):

a. $ \lim_{x \to 0^-} f(x) = 1 $

b. $ \lim_{x \to 0^+} f(x) = -1 $

c. $ \lim_{x \to 3} f(x) = 0 $

d. $ f'(3) $ does not exist

e. $ f'(-1) = 2 $

**Graph Description:**

The graph should demonstrate a function with a jump discontinuity at $ x = 0 $, where the left-hand limit (as $ x \to 0^- $) approaches 1 and the right-hand limit (as $ x \to 0^+ $) approaches -1. As $ x \to 3 $, the function should approach 0, but there should be a point or a cusp to indicate that the derivative at $ x = 3 $ does not exist. At $ x = -1 $, the slope of the function, represented by the derivative, should equal 2, indicating a linear tangent.

**Problem 4: Function Limit Exploration**

Is it possible that there exists some function $ g(x) $ and some constant $ L $ such that

\[
\lim_{x \to 2^-} g(x) = \lim_{x \to 2^+} g(x) = L
\]

but

\[
\lim_{x \to 2} g(x)
\]

does not exist? If so, draw a sketch of such a function. If no, explain why not.

**Explanation:**

A function $ g(x) $ can have the left-hand and right-hand limits equal to a constant $ L $, yet the general limit $ \lim_{x \to 2} g(x) $ might not exist if there is a point of discontinuity (such as a hole or a jump at $ x=2 $). The graph should have left-hand and right-hand limits meeting at $ L $, but a disruption at $ x=2 $.

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