Mmmm! I love apple cider punch this time of year, and so do all my neighbors. My recipe calls for two quarts of apple cider, three
teaspoons of cinnamon, and one pint of lemon-lime soda. This will
make enough for eight people to share. In my neighborhood, however,
there are 36 people. How much do I need of each ingredient to
make enough for my whole neighborhood to share?

Step-by-step explanation:

1/2 cup sugar

1/8 teaspoon ground cinnamon

Pinch table salt

For the doughnuts:

2 1/2 cups all-purpose flour

1 teaspoon baking powder

1/2 teaspoon baking soda

1/2 teaspoon ground cinnamon

1/4 teaspoon ground nutmeg

1/4 teaspoon table salt

1/2 cup thawed apple juice concentrate, at room temperature

1/3 cup sugar

1/4 cup buttermilk

4 tablespoons unsalted butter, melted and cooled

1 large egg

2 quarts vegetable oil, for frying

For the coating: Whisk sugar, cinnamon and salt together in medium bowl; set aside.

For the doughnuts: Whisk flour, baking powder, baking soda, cinnamon, nutmeg and salt together in bowl. Whisk apple juice concentrate, sugar, buttermilk, melted butter and egg together in large bowl. Whisk half of flour mixture into apple juice concentrate mixture until smooth. Add remaining flour mixture; using rubber spatula, use folding motion to mix and press dough until all flour is hydrated and no dry bits remain. (Dough can be covered with plastic wrap and refrigerated for up to 24 hours.)

Dust counter heavily with flour. Turn out dough onto floured counter, then dust top of dough with additional flour. Using your floured hands, gently pat dough into 1/3-inch-thick round, 10 to 11 inches in diameter. Using floured 3-inch round cutter, cut out 9 to 10 doughnut rounds. Using 1-inch round cutter, cut hole in center of each round.

Lightly dust rimmed baking sheet with flour. Transfer doughnut rounds and holes to prepared sheet. Combine dough scraps, then knead into cohesive ball and pat into 1/3-inch-thick round. Cut out 2 or 3 more doughnut rounds and holes (you should have 12 of each). Transfer to sheet and refrigerate while heating oil.

Set wire rack in second rimmed baking sheet and line half of rack with triple layer of paper towels. Add oil to large Dutch oven until it measures about 1 1/2 inches deep and heat over medium-high heat to 350 degrees. Add 6 doughnut rounds and cook, flipping every 30 seconds, until deep golden brown, about 2 minutes. Adjust burner as needed to maintain oil temperature between 325 and 350 degrees.

Using spider skimmer or slotted spoon, transfer doughnuts to paper towel–lined side of prepared rack and let sit while frying remaining doughnut rounds. Return oil to 350 degrees and repeat with remaining doughnut rounds.

Return oil to 350 degrees and, using spider skimmer or slotted spoon, carefully add doughnut holes to hot oil. Cook, stirring often, until deep golden brown, about 2 minutes. Transfer to paper towel–lined side of wire rack. Lightly toss doughnuts and doughnut holes in coating and transfer to unlined side of wire rack. Serve.

not only in mathematics, scientific proof also are required in the other field of science. also maths proofs are of particular structure, containing steps and logics. so proving is like doing science, it is science. that's why it is important to prove. 

an inaccurate proof leads to errors, or stating results that are not accurate.  which wil mess the probelm up you dont want this

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.