[Engineering Maths] Calculus

Question

What can we say about e^-x ?

a. Function is increasing at increasing rate

b. Function is decreasing at increasing rate

c. Function is increasing at decreasing  rate

d. Function is decreasing  at decreasing  rate

Comments

Is it b?

Answer 1

Answer should be D. Function is decreasing at decreasing rate.

Sign of first derivative tells whether a function is increasing or decreasing at particular point.

$\frac{\mathrm{d}}{\mathrm{d} x} e^{-x}= -e^{-x}$

At every point the derivative is negative, hence the function is decreasing.

Second derivative at particular point measures how fast the function is changing at that point.

$\frac{\mathrm{d^2}}{\mathrm{d} x^2} e^{-x}= e^{-x} = \frac{1}{e^x}$

As x increases the rate of change of $e^{-x}$ i.e. $\frac{1}{e^x}$ decreases. Hence the answer D. Function is decreasing at decreasing rate.

Comments

Can you please tell me the behavior for -x^3. Here both derivatives(first and second are negative). As first is -ve so decreasing and second is coming as -6x, means more value of x,less is the change.So can i conclude that -x^3 is decreasing at decreasing rate?

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I see your confusion. When I said $\frac{1}{e^x}$ is decreasing with increasing values of $x$, I should have said the absolute value is decreasing with increasing values of $x$. i.e. the value is going towards zero. The magnitude of second derivative represents the rate of change of rate of change of function. The sign of second derivative tells us whether the value(not absolute) of the slope of function is decreasing or increasing. Means if second derivative is negative that tells us that as x increases the the value of the slope of original function decreases. So if slope was (-100) before it may be (-1000) now. In the case you mentioned $-6x$ is going to negative infinity as $x$ increases. Sure the slope decreases as $x$ increases, but it is becoming more and more steep. So in this case the function is decreasing with increasing rate.

While writing this I realized it's hard for me to explain without pen and paper. You should try plotting different graphs on fooplot.com. Also try following applets from MIT, they should help.

http://mathlets.org/mathlets/creating-the-derivative/
In this one try selecting different functions, and see how $f'$, $f''$ changes as $x$ changes.

http://mathlets.org/mathlets/graph-features/
In this one, try different values to see concavity, convexity and the sign of second derivative.

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Got it now. Absolute value thing cleared all confusions. Thanks :)