# Calculus 1 Project

*label*Mathematics

*timer*Asked: Oct 23rd, 2018

*account_balance_wallet*$35

### Question Description

Hi There Two attached's One is the project guidelines and the other is the functions please select Only One function and work on it

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## Tutor Answer

Here is the solution https://files.fm/u/5wuuxf8mLet me know if anything else is needed.

Contents

1 Function

2

2 Graph of the Function

2

3 Intercepts of the Function

3

4 Graph of the Intercepts of the Function

3

5 Continuity

4

6 Asymptotes

4

7 End behavior of the function

4

8 First and Second Derivatives

5

9 Intercepts of the First Derivative

6

10 Graph of the First Derivative

7

11 Intercepts of the Second Derivative

7

12 Graph of the Second Derivative

8

13 Sign Diagram for First Derivative

9

14 Sign Diagram for Second Derivative

9

15 Discussion

10

A Appendix

11

1

1

Function

The chosen function is f (t)

Equation of f (t) is:

f (t) = 50

t2 + 2t + 4

t2 + 4t + 8

Let D(t) be the denominator of f (t)

D(t) = t2 + 4t + 8 = (t + 2)2 + 4

∴ D(t) 6= 0

∀t ∈ R

Domain = (−∞, ∞)

2

Graph of the Function

2

!

3

Intercepts of the Function

The following MATLAB code computes the x and y intercepts of f (t)

1

f = @(t) 50*(t.ˆ2+2*t+4)./(t.ˆ2+4*t+8);

2

3

a = solve(f);

4

a =a(imag(a)==0);

5

b = f(0);

6

7

v =['The x intercept is',num2str(a)];

8

v0 = ['The y intercept is ' ,num2str(b)];

9

disp(v);

10

disp(v0);

Command Window

∴ There is no x intercept, the y intercept is 25

Point of interception is (0, 25)

4

Graph of the Intercepts of the Function

3

5

Continuity

D(t) = t2 + 4t + 8 = (t + 2)2 + 4

∴ D(t) 6= 0

∀t ∈ R

∴ f (t) is continuous on the interval (−∞, ∞)

6

→ No discontinuities exist

Asymptotes

D(t) 6= 0

∀t ∈ R

→ No vertical asymptotes

To find the horizontal asymptote ha :

ha = limt→∞ f (t) = 50 × limt→∞

ha = 50 ×

t2 + 2t + 4

t2 + 4t + 8

!

2

4

t 1 + t + t2

= 50 × limt→∞

8

4

2

t 1+ + 2

t t

1+0+0

= 50

1+0+0

A horizontal asymptote exists at y = 50

7

End behavior of the function

It was calculated during finding the horizontal asymptote of f (t)

limt→∞ f ...

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