Calculus Assignment Help: Temperature & Daylight Equations Explained

[PatrickrG]

Homework Statement

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1. When a certain object is placed in an oven at 540°C, its temperature T(t) rises according to the equation T(t) = 540(1 – e^–0.1t), where t is the elapsed time (in minutes).

What is the temperature after 10 minutes and how quickly is it rising at this time?

I have come to the conclusion that when plugging 10 into t I get 341.35 as the temperature, but I am unsure how to find out how quickly it is rising?

2.
The amount of daylight a particular location on Earth receives on a given day of the year can be modeled by a sinusoidal function. The amount of daylight that Windsor, Ontario will experience in 2007 can be modeled by the function D(t) = 12.18 + 3.1 sin(0.017t – 1.376), where tis the number of days since the start of the year.

a) On January 1, how many hours of daylight does Windsor receive?

b) What would the slope of this curve represent?

c) The summer solstice is the day on which the maximum amount of daylight will occur. On what day of the year would this occur?

d) Verify this fact using the derivative.

e) What is the maximum amount of daylight Windsor receives?

f) What is the least amount of daylight Windsor receives?

I have found for a that it would be 9.15 hours of sun and that b would be the daily change in sunlight
, I am stuck with c through f :(

2. Homework Equations
T(t) = 540(1 – e^–0.1t)
D(t) = 12.18 + 3.1 sin(0.017t – 1.376),

The Attempt at a Solution

I have attempted some but the others unsure how to...[/B]

 

[member 587159]

1) What does T'(t) mean?
2) I suppose this can be done with a graphing calculator, as d) asks to verify it with the derivative.
Anyway, what do you know when f'(x) = 0?

 

[PatrickrG]

1) What does T'(t) mean?
2) I suppose this can be done with a graphing calculator, as d) asks to verify it with the derivative.
Anyway, what do you know when f'(x) = 0?

T'(t) ? would that be its derivative?
I do not have a graphing calculator, and I am not sure what it means when f'(x) is zero

 

[member 587159]

T'(t) ? would that be its derivative?
I do not have a graphing calculator, and I am not sure what it means when f'(x) is zero

1) Yes it's the derivative

$$T'(t) = dT(t)/dt$$

2) If f'(x) is zero and the sign of f'(x) changes where f'(x) is zero, f reaches an extremum. I'm quite sure your textbook mentions this.

 

[PatrickrG]

so for 1 I just need to find the derivative to know the temperature rise?
and for 2 if the derivative is zero and the sign changes it will make a max or a min? Sorry I still don't know how to go about this when it comes to the formula or the numbers

 

[member 587159]

so for 1 I just need to find the derivative to know the temperature rise?
and for 2 if the derivative is zero and the sign changes it will make a max or a min? Sorry I still don't know how to go about this when it comes to the formula or the numbers

I recommend you to learn the theory before you start making exercises.

1) You have to find the derivative and find where it reaches a maximum/minimum. You do this by solving f'(x) = 0 for x. The x-value you get, x1, will be the x-value of the maximum. The corresponding maximum value is given by f(x1).
2) See 1)

The derivative of a function in a point shows the slope of the tangent line in that point. Intuitively, you know that the slope of the tangent at an extremum is zero, since the tangent line in a maxium is a line parallel to the x-axis. That's why you solve f'(x) = 0 to find the maximum/minimum.