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
Well, congratulations on your sketches! It sounds like you've got the right idea. For a curve with a positive and increasing slope, imagine a smiley face climbing up a hill. The curve should be concave up, meaning it looks like a frown turned upside down. As for the equation, you could try something like y = x^2. This is a simple parabola that opens upward.
Now, for a curve with a positive and decreasing slope, picture a sad face sliding down a hill. The curve should be concave down, like a regular frown. For the equation, you could use y = -x^2. This is just the same parabola as before, but flipped upside down.
Remember, these are just examples to help you understand the concept. Feel free to get creative and experiment with different shapes and equations!
Yes, you correctly drew the curves for parts a and b. In part a, a curve with a positive slope that is increasing will have a concave up shape, meaning it curves upward as you move along the curve. In part b, a curve with a positive slope that is decreasing will have a concave down shape, meaning it curves downward as you move along the curve. These shapes indicate the changing steepness of the slope.
For part c, here are examples of equations that represent curves with these properties:
a) To represent a curve with a positive slope that is increasing, you can use a quadratic equation. One example is y = x^2, which is a simple parabolic curve with a positive slope that increases as you move along the curve.
b) To represent a curve with a positive slope that is decreasing, you can use an exponential equation. One example is y = e^(-x), which is a decaying exponential curve. It starts with a positive slope and gradually decreases as you move along the curve.
These are just a few examples, and there are many other equations that can represent curves with positive slopes that are increasing or decreasing.
Yes, you did a good job in sketching the slopes for both cases.
a) To sketch a curve with a positive and increasing slope, you can consider an equation of the form y = f(x), where f'(x) > 0 and f''(x) > 0. This means that the first derivative is positive and is increasing, and the second derivative is positive. For example, you can consider the equation y = x^3, which has a positive and increasing slope throughout its domain.
b) To sketch a curve with a positive and decreasing slope, you can consider an equation of the form y = g(x), where g'(x) > 0 and g''(x) < 0. This means that the first derivative is positive and is decreasing, and the second derivative is negative. For example, you can consider the equation y = -x^3, which has a positive and decreasing slope throughout its domain.
In both cases, the equations of the curves correspond to power functions. For a curve with a positive and increasing slope, it generally has a concave up shape, while a curve with a positive and decreasing slope generally has a concave down shape.
Remember that the choice of equations is not limited to the examples provided above, and you can explore different functions to achieve the desired properties.