calculus

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!

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  1. (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

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