Here’s a short and sweet Micro Softy for this week.

Two days ago, little Kaylee celebrated her 5^th birthday. Interestingly, next year she will turn seven years old.

In the absence of time warps and worm holes, explain how this seemingly weird case of growing older faster can make sense.

Now for last week’s tougher one:

Solution to Micro Softy 32:  Elementary Calculus is Inconsistent!

Last week’s Micro Softy teased an apparent inconsistency in elementary calculus, as follows: We know (d/dn) n² = 2n. But n² = n+n+n+…+n. There are n terms. It follows that

(d/dn) n² = (d/dn) [n+n+n+…+n] = 1+1+1+…+1 = n.

This is a different answer! What’s wrong here?

The answer is that derivatives only apply to continuous (smooth) functions. The first derivative is the slope of a tangent line. Take a look at the plot of n² shown.  The ideal of a slope doesn’t even apply.

The step n² = n+n+n+…+n says n is an integer and eliminates any idea of a continuous function:

(The Micro Softy is a weekly feature of Mind Matters News. Links to all the previous ones are available through this one.)

Monday Micro Softy 32: Elementary calculus is inconsistent! This is a fascinating puzzle — one that usually stumps even my best graduate students. The solution to Micro Softy 31 is to work back from the end of the alphabet. You will notice a term you can easily solve.

Image Credit: MarekPhotoDesign.com - Adobe Stock

Monday Micro Softy 31: Elementary algebra. How much can an equation be simplified? In last week’s puzzle, the car was handling like a cement mixer because something was wrong with the tires. The solution lies in finding out what that is.

Monday Micro Softy 30: Driving a cement mixer. Why did the car with leaky tires suddenly start handling like a cement mixer, and then, just as suddenly, handle normally again? Micro Softy 29 shows that unambiguous facts can be stated in confusing language but cold hard logic can solve the puzzle.

Links to all the previous Monday Micro Softies

Monday Micro Softy 29: A funeral lament in four lines. The funeral director was puzzled by Dan’s description of his relationship to the deceased but there was no question that his grief was sincere Here, you will also find links to Microsofties 22 through 29 as well.

Monday Micro Softy 21: Finding More of the Deadly Fentanyl Pills. There, you will also find links to Microsofties 11 through 20 as well.

At Monday Micro Softy 11: What Happened to That Other Dollar?, you will find links to the first ten Micro Softies. Have fun!