Differentiation with constants: U(t) = A(1 - e^{-Bt^2}) + c; A,B,c > 0

[P0ppy]

Hello! As the title suggests I need to differentiate a function with a couple of constants. I'm from Denmark but I'll do my best to translate the problem:

A field receives fertilizer. The yield of a crop (measured in metric tons per hectare) at the addition of t ≥ 0 units of fertilizer per hectare is referred to as U(t). As a model for this function we use:

\(\displaystyle U(t)\, =\, A\left(1\, -\, e^{-Bt^2}\right)\, +\, c,\)

where A, B, and C are positive constants.

Determine U'(t) and use it to figure out whether U(t) is an increasing or decreasing function of t ≥ 0.

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Thanks in advance for any help! If my translation is questionable or my wording was confusing in any way, please don't hesitate to point it out.

 

[tkhunny]

Hello! As the title suggests I need to differentiate a function with a couple of constants. I'm from Denmark but I'll do my best to translate the problem:

A field receives fertilizer. The yield of a crop (measured in metric tons per hectare) at the addition of t ≥ 0 units of fertilizer per hectare is referred to as U(t). As a model for this function we use:

\(\displaystyle U(t)\, =\, A\left(1\, -\, e^{-Bt^2}\right)\, +\, c,\)

where A, B, and C are positive constants.

Determine U'(t) and use it to figure out whether U(t) is an increasing or decreasing function of t ≥ 0.

​

Thanks in advance for any help! If my translation is questionable or my wording was confusing in any way, please don't hesitate to point it out.

What have you tried? How do you think it works? Is there a difference between ADDED constants and MULTIPLIED constants?

For any constant, c, \(\displaystyle \dfrac{d}{dt}c = 0\)


For any constant, A, \(\displaystyle \dfrac{d}{dt}\left(A\cdot f(t)\right) = A\cdot f'(t)\)

Let's see where that leads.

 

[P0ppy]

Is this correct?

 

[Dr.Peterson]

Is this correct?

You're missing a negative, and you didn't completely apply the chain rule. Notice that the exponent is not just -Bt, but -Bt^2.