INPUT ARTICLE: Article: Place 3 cups of blueberries, 3 tablespoon (44.4ml) of lemon juice, 3/4 cups of sugar, 1/4 tsp. of salt, and 1/8 tsp. of cinnamon in a medium saucepan. Continue to stir the ingredients to incorporate them. Do this while the mixture is heating. When you're done, let the ingredients cool for 10 minutes. Stir in 1 1/2 cups of full fat plain yogurt and 1/2 cup of whole milk into the ingredients until they're completely incorporated. Check the instructions on the ice cream maker, as this time may vary. You can enjoy this frozen yogurt right away or freeze it for a few hours first.

SUMMARY: Place the blueberries, lemon juice, sugar, salt, and cinnamon in a medium saucepan. Heat the ingredients over medium heat until the sugar has dissolved. Mash up the blueberries with a potato masher. Stir the yogurt and milk into the mixture. Chill the mixture in the refrigerator until it's completely cold. Process the blueberry yogurt mixture in your ice cream maker for 25 minutes. Serve.

INPUT ARTICLE: Article: The dough will be rectangular. Arrange the breadsticks on a baking sheet.

SUMMARY: Mix dry ingredients together and then add liquids. Mix them well. Allow the dough to rise. Preheat an oven to 350F/180C. Take out the dough from the dish and place it down on a cutting board. Roll out the dough. Cut the dough, so you have each "stick" 1/2-3/4 inches (1.2-2cm) deep. Cut the dough into sticks measuring approximately 1 by 4 inches (2.5cm x 10cm) in diameter. Combine all the seasoning toppings in another bowl. Apply the seasoning toppings to the sticks. Bake the sticks for 12-15 minutes at the temperature specified above. Finish making the breadsticks. Serve hot.

INPUT ARTICLE: Article: Being single is associated with a decreased risk of adverse health effects from dysfunctional marriages or intimate relationships. Some studies have shown that bad relationships can actually make you physically unhealthy. But singles are more likely to be fit and healthy. Singles also tend to go to the gym more, eat better, and have lower stress levels. Single people tend to be thinner than people who are in a relationship. One study found that people gain an average of 14 or more pounds after starting a relationship. Single people also get more, good quality sleep than people who share a bed with a partner. Single women have better mental health than married women, especially married women who have children.

SUMMARY: Consider the health benefits of being single.

INPUT ARTICLE: Article: Higher-level speed calculations can get confusing because mathematicians and scientists use different definitions for "speed" and "velocity". A velocity has two components: a magnitude and a direction. The magnitude is equal to the object's speed. A change in the direction will cause a change in the velocity, but not in the speed. For example, let's say that there are two cars moving in opposite directions. Both cars' speedometers read 50 km/hr, so they both have the same speed. However, since they are moving apart from each other, we say that one car has a velocity of -50 km/hr and one has a velocity of 50 km/hr. Just as you can calculate instantaneous speed, you can also calculate instantaneous velocity. Objects can have velocities with a negative magnitude (if they are moving in a negative direction relative to something else). However, there's no such thing as a negative speed, so in these cases the absolute value of the magnitude gives the object's speed. For this reason, in the example problem above, both cars have a speed of 50 km/hr. If you have a function s(t) that gives you the position of an object with regards to time, the derivative of s(t) will give you its velocity with regards to time. Just plug a time value into this equation for the variable t (or whatever the time value is) to get the velocity at this given time. From here, finding the speed is easy. For example, let's say that an object's position in meters is given with the equation 3t2 + t - 4 where t = time in seconds. We want to know what the speed of the object is at t = 4 seconds. In this case, we can solve like this: 3t2 + t - 4 s'(t) = 2 3t + 1 s'(t) = 6t + 1 Now, we plug in t = 4: s'(t) = 6(4) + 1 = 24 + 1 = 25 meters/second. This is technically a velocity measurement, but since it's positive and direction is not mentioned in the problem, we can essentially use it for speed. Acceleration is a way of measuring the change in an object's velocity over time. This topic is a little too complex to explain fully in this article. However, it's useful to note that when you have a function a(t) that gives acceleration with regards to time, the integral of a(t) will give you velocity with regards to time. Note that it's helpful to know the object's initial velocity so that you can define the constant that results from an indefinite integral. For example, let's say that an object has a constant acceleration (in m/s2 given by a(t) = -30. Let's also say that it has an initial velocity of 10 m/s. We need to find its speed at t = 12 s. In this case, we can solve like this: a(t) = -30 v(t)= a(t)dt = -30dt = -30t + C To find C, we'll solve v(t) for t = 0. Remember that the object's initial velocity is 10 m/s. v(0) = 10 = -30(0) + C 10 = C, so v(t) = -30t + 10 Now, we can just plug in t = 12 seconds. v(12) = -30(12) + 10 = -360 + 10 = -350. Since speed is the absolute value of velocity, the object's speed is 350 meters/second.

SUMMARY:
Understand that speed is defined as the magnitude of velocity. Use absolute values for negative velocities. Take the derivative of a position function. Take the integral of an acceleration function.