Posted by Travis on .
a) Sketch a curve whose slope is always positive and increasing.
b) Sketch a curve whose slope is always positive and decreasing.
c) Give equations for curves with these properties.
So for a, I drew a positive slope that was concave up. And for b, I drew a slope that was concave down. I drew both of these slopes going in a upward/right direction. Did I do that correctly? Also, I'm not sure how an equation should look for this? Thanks!
(a) y = e^x, since y'' = y' = e^x (both positive)
(b) y = -e^-x, since
y' = e^-x positive
y'' = -e^-x negative
y = arctan(x) also fits here, since
y' = 1/(1+x^2) positive
y'' = -2x/(1+x^2)^2 negative
So the curve can change from concave up to concave down and still have decreasing slope