The Flux Capacitor 4 great activities to charge your calculus class Anne Barber anne.barber@wrps.net Karen Scarseth Karen.scarseth@wrps.net Green Lake, May 5 th, 2017 Piece O-Cake Glass Half Full Relating the Rates Cake Bake-Off
Piece - O Cake Find the total volume of the cake from one piece. 1) Trace the outline of the piece of cake on centimeter graph paper. Place the cake so the center (x-axis) to the peak is 7.5cm. 2) Split outline of cake into two functions. Collect coordinates of points for each function (at least ten per function). 3) Enter coordinates into the STAT (Lists) functions of your calculator and find a regression function that fits your data. 4) Use the two functions and the Washer method to find the volume of the cake. (Your piece is rotated around the x-axis to form a bundt cake.) 5) You must turn in the following: I. Graph with all points identified and both functions labeled. II. Table of data points for both functions. III. State each function. (You may round to thousandths place when you write it, but don t round the functions in your calculations.) IV. Integral set up with limits. V. Final answer with label.
Glass Half Full You will find from home a glass or vase that will hold liquid. The best type of glass is a wine glass. For more than basic credit, you wish to stay away from a glass with straight edges. The more complicated the glass, the better this assignment will work. Simple Glass too easy Still Simple but better Types of glass you want Your job is to find 3 pieces of information 1. Find the volume that the object will hold 2. Find the inner surface area of the glass 3. If the water is being poured at the rate of 16 cubic cm per second, at what rate is the water level rising when the glass contains half of its volume? First, you may NOT use a measuring cup to determine the volume. You must generate equations to model the shape of the glass. For the middle glass above, you can generate the side of the glass using a line. To model the glasses on the right, you will need a piecewise function. I will not tell you how to generate equations for the graph. However, somehow, you need to determine the radius of the glass as regularly spaced height intervals. You may use regression on your calculator to determine the equation. Once you generate your equations, you need to flip it about the (easier) or to generate a solid. Then calculate the volume in cubic centimeters. Then you will use a measuring cup to measure the volume and determine the difference between the actual volume and the volume generated using your equation(s). To find the surface area, you will use the inner surface area since some glasses have a fluted outer edge which would complicate the outer surface calculations. To find the related rate, you need to determine the rate of change of the volume.
What you will hand in: The glass (which I will return to you). Make sure your name is taped to the inside of it. The glass should not be made of crystal. It should not be expensive. You may want to go to a store and purchase a cheap plastic glass to your liking. A paragraph describing how you generated your equations. Be sure that you describe what you did. If you generated a table of values, show that table and how you determined them. The equations you generated for one side of the glass. The formula that you used for the volume and the answers that you found. The true volume of the glass and the difference between the true area and the volume you found with your equations. The analysis of the related rates problem and the answer. How you will be graded: The assignment will be worth 20 points times the difficulty level of the glass (which is determined by me) A very simple glass has a difficulty level 1 maximum 20 points A glass with straight sides (not vertical) has difficulty level 1.25 maximum 25 points A glass with a linear piecewise function has difficulty level 1.5 maximum 30 points A glass with more than 2 pieces and/or includes non-linear sides has difficulty level 2 maximum 40 points This project will be weighted in the grade book to be a 50 point assignment.
Relating the Rates A Related Rates Activity In a group of 2-3 students, complete the activity below. It will count as a 10 point summative grade. You will only have one chance to turn in your final answers. Show your final answers to me when you are done. All measurements should be done in cm. Part 1: Pick a cone. Which one did you pick? Measurements of the cone: Height Diameter Radius If the cone was filled with sand that began falling out the bottom, determine the rate of change of the volume if the height is changing at a rate of -1.15cm/sec and the radius is changing at a rate of -0.35cm/sec at the instant the height of the sand is half the original height. dv/dt = Part 2: Pick a rectangular prism. Which one did you choose? Measurements of the rectangular prism: Length Width Height If the sand from the cone is falling directly into your rectangular prism, what is the rate of change of the height of the sand in your prism? dh/dt = Part 3: Pick a cylinder. Which one did you choose? Measurements of the cylinder: Height Diameter Radius The rectangular prism has a hole on a lower corner and the sand is falling out at twice the rate it filled the prism. It is falling directly into the cylinder. What is the rate of change of the height of the sand in the cylinder? dh/dt =
Extension: A breakdown of an AP question. Consider a conical tank whose radius at the top is 4 feet and whose depth is 10 feet. It s being filled with water at the rate of 2 ft 3 /min. How fast is the water level rising when it is at a depth of 5 feet? Start by answering the questions below. A) What is the main geometric formula you need? B) What is the derivative of formula from part A, with respect to time? C) Analyze your derivative in part B. What variables do you have? Which do you need? Which one answers the original question? D) Assuming A, B, and C are correct, you don t have enough information to answer the question the way you took the derivative. Start again with the formula from part A. You need to eliminate one of the variables. Which one should you try to eliminate? (Look at what you were missing in part C.) E) Now use the initial information of the radius, the height, and the specific height of the water. Set up a proportion using these three measurements a variable for the radius of the water. F) Rewrite the formula from part A by substituting in an equivalent form of r. G) Finish by taking the derivative of the formula from part F and substituting the values you have for the remaining variables. Give a final answer to the original question with a correct label.
Cake Bake-Off --End of year Calculus Project Hurrah! It s the end of the year and we need to celebrate! Let s have some cake. Break out your aprons because we are going to be in the kitchen!! This project has 3 parts to complete. You will write calculus problems, solve calculus problems, and bake a cake. That s it of course we will eat some cake too. First, you need to find a group to work with. Your work must remain a secret and cannot be shared with the other groups. We need five groups. Each group can have 4 to 5 people. You will all sign a contract agreeing to the grade your group receives. PART 1: Write the problems You will be given a set of answers. You must NOT tell any other group what you have. For each answer, you need to write a calculus problem from what you have learned this year. Please see the list of types of problems you need to include in your work. It needs to be neat and easy to follow. You may type or hand-write the problems. AGAIN, don t share your work with the other groups. PART 2: Solve the problems I will exchange the problems with other groups. You will then need to solve the calculus problems. The solutions will be the missing values of the recipes. No help from any teacher will be given. There should be supporting work for all the problems. You will need to turn in a copy of your completed work and answers. You will need to keep a copy of your answers for part 3. AGAIN, don t share your work or answers to the problems you wrote with the other groups. PART 3: BAKE We will then take your answers and put them into the recipes for cakes, fillings, and frostings. We will spend a few days in the kitchen making the recipes. AND THEN We will have a cake contest. You will be judge on presentation, taste, and math. SO, take your time on the calculus. If you did something wrong, we will taste it. Please review the each part of the project so you understand how you are graded. Please review the deadlines for each part and let me and your group know if you will be out of class during any part of the project. Have Fun!! Part one: Due Part two: Due Part three: Kitchen
PART ONE You need to write calculus problems that will result in the list of answers given to you. Make sure you include at least one of each of the following types of problems. Number the problems the same way as the answers are listed. You can NEATLY write or type the problems. Types of Problems: Solving Limit from graph L Hopital Limit Derivatives at specific point that include trig rules, implicit, chain rule Related Rate Max/Min problem Graph of derivative and finding max/min/poi of original function Definite integrals including trig rules, exponential, natural logs Exponential Growth/Decay Newton Law of Cooling Area under a curve Volume of a rotation Volume of a cross section You will be graded on the variety, neatness, level of difficulty, and correct mathematics.
CALCULUS CAKE A 1. 13 2. 1⅛ 3. 3 4. 2.25 5. 1.5 6. 0.25 7. 1 8. 2 9. 2 10. 0.5 11. 1 12. ⅛ 13. 0.75 14. 350 15. 2 16. 9 17. 33 18. 1.5 19. 6 20. 0.25 21. 2 22. 1 CALCULUS CAKE B 1. 2.25 2. 1 3. 6 4. 2 5. 1 6. 1.75 7. 4 8. 1 9. 12 10. 350 11. 2 12. 9 13. 0.5 14. 1.5 15. 0.5 16. 25 17. 2.25 18. 0.25 19. 6 20. 1.5 21. 9 22. 0.75
CALCULUS CAKE C 1. 0.75 2. 2 3. 3 4. 1.5 5. 1.5 6. 3 7. 1.5 8. 1.5 9. 0.75 10. 1 11. 350 12. 3 13. 9 14. 25 15. 30 16. 10 17. 2.25 18. 0.25 19. 6 20. 1.5 21. 9 22. 0.75 CALCULUS CAKE D 1. 2 2. ⅔ 3. 2 4. 1 5. 0.5 6. 2 7. 2 8. 4 9. 2 10. 2 11. 350 12. 2 13. 9 14. 35 15. 40 16. 8 17. 1 18. ⅛ 19. 1 20. 1.5 21. 0.25 22. 3.5 23. 2 24. 30
CALCULUS CAKE E 1. 0.75 2. 2 3. 1 4. 1 5. 2.5 6. 1.5 7. 2 8. 2 9. 2 10. 1 11. 1 12. 1 13. 350 14. 2 15. 9 16. 22 17. 28 18. 4 19. 2 20. 1.5 21. 0.5 22. 1 23. 2
Ingredients: Calculus Cake #1C 1 cups butter, softened 2 cups sugar 3 eggs cups mashed ripe bananas 4 5 teaspoons vanilla extract 36 cups all-purpose flour 7 8 9 teaspoons baking powder teaspoons baking soda teaspoon salt 10 cups buttermilk Preparation: Preheat oven to 350. 11 Generously grease and line with parchment paper three 12 9-13 in. round baking pans. In a large bowl, cream butter and sugar until light and fluffy. Add eggs, one at a time, beating well after each addition. Beat in bananas and vanilla. Combine the flour, baking powder, baking soda and salt; Add to creamed mixture alternately with buttermilk, beating well after each addition. Bake for 14-15 minutes or until a toothpick inserted near the center comes out clean. Cool for 1016 minutes before removing from pans to wire racks to cool completely.
Frosting and Filling Ingredients: 2¼ 17 cups powdered sugar ¼ 18 cup unsweetened cocoa powder Pinch of Salt 619 oz cream cheese 1½ 20 sticks unsalted butter, softened 921 oz bittersweet chocolate, melted and slightly cooled ¾ 22 cup sour cream Preparation 1. Sift together sugar, cocoa, and salt. 2. Beat cream cheese and butter with mixer on medium-high speed until smooth. Reduce speed to medium low; gradually add sugar-cocoa mixture, and beat until combined. Pour in chocolate in a slow steady stream. Add sour cream; beat until combined. Putting it all together: Place one layer of cake on serving dish. Cover the layer with half the frosting. Place the second layer on top. Cover the top with the remaining frosting.
Calculus Cake #2D For cake: 11 cup all-purpose flour 21/3 cup unsweetened Dutch-process cocoa powder 31 teaspoon baking soda 41 teaspoon baking powder 5 teaspoon salt 61 stick unsalted butter, softened 71 cup packed light brown sugar 82 large eggs at room temperature 30 minutes 91 teaspoon pure vanilla extract 10 1 cup whole milk Make cake: Preheat oven to 350 11 F with rack in middle. Butter bottom and side of 212, 913 -in round pans, then line bottoms with a round of parchment. Sift together flour, cocoa powder, baking soda, baking powder, and salt into a bowl. Beat together butter and brown sugar in a large bowl with an electric mixer at medium-high speed until light and fluffy, 3 to 5 minutes. Add eggs 1 at a time, beating well after each addition, then beat in vanilla. Reduce speed to medium-low and add flour mixture and milk alternately in batches, beginning and ending with flour mixture (batter will appear curdled). Transfer batter to cake pans, even amount in each, and smooth tops. Bake until cake begins to pull away from side of pan and a wooden pick inserted into center comes out clean, 3514 to 4015 minutes. Cool cakes in pan 5 minutes, then invert onto a rack and cool completely, about 1 hour.
Make Glaze FIRST, as it needs to cool for several minutes. For filling: 16 8 ounces, weight Cream Cheese, Softened 17 1 18 ⅛ 19 1 cup Powdered Sugar teaspoons Salt teaspoon Vanilla 20 1-½ cup Heavy Whipping Cream Make filling: Blend whipping cream on high in bowl until light and fluffy. Set aside. In another bowl, mix together on medium speed cream cheese, sugar, salt and vanilla. Fold in whipped cream. For glaze: 21 22 1/4 cup heavy cream 3 1/2 oz bittersweet chocolate finely chopped 232 teaspoons light corn syrup Make glaze: Bring cream to a simmer in a small heavy saucepan over medium heat, then pour over chocolate in a bowl and let stand 1 minute. Gently whisk until smooth, then stir in corn syrup. Cool completely, gently stirring occasionally, about 30 24 minutes (glaze will thicken). Put it all together: Put cake together by positioning one cake layer on a plate. Then spread on filling. Place second layer on top. Pour glaze over cake.
Calculus Cake #3B Ingredients: 21 1/4 cups cake flour, plus more for dusting the pans 12 cups whole milk, at room temperature 63 large egg whites, at room temperature 24 teaspoons almond extract 15 teaspoon vanilla extract 16 3/4 cups granulated sugar 47 teaspoons baking powder 18 teaspoon table salt 12 9 tablespoons unsalted butter, softened but still cool Preparation 1. Heat oven to 350 10 degrees. Spray two 11 129 -inch round cake pans with nonstick cooking spray; line the bottoms with parchment or waxed paper rounds. Spray the paper rounds, dust the pans with flour, and invert pans and rap sharply to remove excess flour. 2. Pour milk, egg whites, and extracts into 2-cup glass measure, and mix with fork until blended. 3. Mix cake flour, sugar, baking powder, and salt in bowl of electric mixer at slow speed. Add butter; continue beating at slow speed until mixture resembles moist crumbs, with no powdery streaks remaining. 4. Add all but 1/2 13 cup of milk mixture to crumbs and beat at medium speed (or high speed if using handheld mixer) for 1 14 1/2 minutes. Add remaining 1/2 15 cup of milk mixture and beat 30 seconds more. Stop mixer and scrape sides of bowl. Return mixer to medium (or high) speed and beat 20 seconds longer. 5. Divide batter evenly between two prepared cake pans; using rubber spatula, spread batter to pan walls and smooth tops. 6. Bake until thin skewer or toothpick inserted in the center comes out clean, 25 16 minutes.
7. Let cakes rest in pans for 3 minutes. Loosen from sides of pans with a knife, if necessary, and invert onto wire racks. Let cool completely, about 1 1/2 hours. Frosting and Filling Ingredients: 2¼ 17 cups powdered sugar ¼ 18 cup unsweetened cocoa powder Pinch of Salt 619 oz cream cheese 1½ 20 sticks unsalted butter, softened 921 oz bittersweet chocolate, melted and slightly cooled ¾ 22 cup sour cream Preparation 3. Sift together sugar, cocoa, and salt. 4. Beat cream cheese and butter with mixer on medium-high speed until smooth. Reduce speed to medium low; gradually add sugar-cocoa mixture, and beat until combined. Pour in chocolate in a slow steady stream. Add sour cream; beat until combined. Putting it all together: Place one layer of cake on serving dish. Cover the layer with half the frosting. Place the second layer on top. Cover the top with the remaining frosting.
13 1 tablespoons butter 12 1/8 cups sugar 3 eggs 24 1/4 cups flour 15 1/2 teaspoons baking powder 1/4 6 teaspoon salt Calculus Cake #4A 17 tablespoon + 82 teaspoons orange juice concentrate 29 tablespoons water 1/2 10 cup plus 11 1 teaspoon milk 1/8 12 cup canola oil ¾ 13 teaspoon vanilla extract Zest from one orange Directions: 1. Preheat oven to 350 14 F. Spray two 15 16 9- inch round cake pans with nonstick cooking spray; line the bottoms with parchment or waxed paper rounds. Spray the paper rounds, dust the pans with flour, and invert pans and rap sharply to remove excess flour. 2. Cream together butter and sugar until light and fluffy. 3. Add eggs, one at a time, beating well after each addition. 4. Sift together flour, baking powder and salt. 5. Combine the orange juice concentrate, water, milk, canola oil, vanilla and orange extract. 6. Add dry ingredients and wet ingredients, alternately, to butter mixture, beating well after each addition. 7. Divide batter among cake pans and bake for 33 17 minutes or until toothpick inserted in center of cake comes out clean. 8. Remove from oven, cool, and remove from pans.
Frosting and Filling Ingredients: 1½ 18 619 Pinch of Salt cups sugar large eggs ¼ 20 21 122 teaspoon cream of tarter cups unsalted butter, cut into tablespoons teaspoon pure vanilla extract Preparation: 1. Heat sugar and eggs; Bring saucepan filled with about 2 inches of water to a simmer. Meanwhile, combine sugar, egg whites, and salt in a large bowl. Set bowl over (not in) simmering water, and whisk until whites are warm to the touch and sugar is dissolved, 2 to 3 minutes. Test by rubbing between your fingers; mixture should feel smooth. 2. Whip meringue: Use electric mixer to beat on low speed until foamy (using whisk attachment). Add cream of tartar, and beat on medium-high speed until stiff, glossy peaks form and mixture is cooled completely. About 10 minutes. 3. Mix in butter: Reduce speed to medium-low; add butter 2 tablespoons at a time, beating to incorporate fully after each addition. Don t worry if the buttercream appears curdled at this point; it will become perfectly smooth again with continued beating. After all the butter has been added, add the vanilla. 4. Finish: Switch to a paddle attachment on the mixer and beat on lowest speed to reduce air bubbles, 3 to 5 minutes. Putting it all together: Place one layer of cake on serving dish. Cover the layer with half the meringue. Place the second layer on top. Cover the top with the remaining meringue.
Calculus Cake #5E Ingredients: 1¾ cup butter softened 2 large eggs 13 teaspoon vanilla 14 teaspoon salt 25 1/2 cups all-purpose flour 16 1/2 cups sugar 2 7 tablespoons red gel food coloring plus 82 tablespoons water 29 tablespoons unsweetened cocoa 110 cup buttermilk 11 tablespoon vinegar 112 teaspoon baking soda Preparation: Preheat oven to 350 13. Generously grease and line with parchment paper two 14 915 -inch cake pans Cream butter and sugar until light and fluffy. Add eggs and beat 122 minute longer. Beat in vanilla until blended. Combine cocoa, food coloring, and 223 tablespoons water; beat into the egg creamed mixture. Combine flour and salt. Alternate adding flour and buttermilk mixtures to the batter, beating on low to medium speed. Combine soda and vinegar in cup and add to cake batter. Spoon batter evenly into the cake pans; bake for 22 16 to 17 28 minutes. Cool on cake racks.
Frosting and Filling Ingredients: 18 19 20 21 Oz. Cream cheese Teaspoons Milk Cups Whipping Cream Cups Powdered Sugar Preparation: In chilled large bowl, mix cream cheese and milk until smooth. Beat in whipping cream and powdered sugar with electric mixer on high speed, scraping bowl occasionally, until soft peaks form. Putting it all together: Place one layer of cake on serving dish. Cover the layer with half the cream cheese frosting. Place the second layer on top. Cover the top with the remaining cream cheese frosting.
Ingredients: Banana Cake Calculus Cake #1C ¾ cup butter, softened (12 Tablespoons) 2 cups sugar 3 eggs 1-1/2 cups mashed ripe bananas (about 3 medium) 1-1/2 teaspoons vanilla extract 3 cups all-purpose flour 1-1/2 teaspoons baking powder 1-1/2 teaspoons baking soda 3/4 teaspoon salt 1 cup buttermilk Preparation: Preheat oven to 350. Generously grease and line with parchment paper three 9-in. round baking pans. In a large bowl, cream butter and sugar until light and fluffy. Add eggs, one at a time, beating well after each addition. Beat in bananas and vanilla. Combine the flour, baking powder, baking soda and salt; Add to creamed mixture alternately with buttermilk, beating well after each addition. Bake for 25-30 minutes or until a toothpick inserted near the center comes out clean. Cool for 10 minutes before removing from pans to wire racks to cool completely. Frosting and Filling Ingredients: 2¼ cups powdered sugar ¼ cup unsweetened cocoa powder Pinch of Salt 6 oz cream cheese 1½ sticks unsalted butter, softened 9 oz bittersweet chocolate, melted and slightly cooled ¾ cup sour cream Preparation Sift together sugar, cocoa, and salt. Beat cream cheese and butter with mixer on mediumhigh speed until smooth. Reduce speed to medium low; gradually add sugar-cocoa mixture, and beat until combined. Pour in chocolate in a slow steady stream. Add sour cream; beat until combined. Putting it all together: Place one layer of cake on serving dish. Cover the layer with half the frosting. Place the second layer on top. Cover the top with the remaining frosting.
Chocolate Cake - Calculus Cake #2D For cake: 1 cup all-purpose flour 1/3 cup unsweetened Dutch-process cocoa powder 1 teaspoon baking soda 1/2 teaspoon baking powder 1/4 teaspoon salt 1 stick unsalted butter, softened 1 cup packed light brown sugar 2 large eggs at room temperature 30 minutes 1 teaspoon pure vanilla extract 1 cup whole milk Make cake: Preheat oven to 350 F with rack in middle. Butter bottom and side of 2, 9-in round pans, then line bottoms with a round of parchment. Sift together flour, cocoa powder, baking soda, baking powder, and salt into a bowl. Beat together butter and brown sugar in a large bowl with an electric mixer at medium-high speed until light and fluffy, 3 to 5 minutes. Add eggs 1 at a time, beating well after each addition, then beat in vanilla. Reduce speed to medium-low and add flour mixture and milk alternately in batches, beginning and ending with flour mixture (batter will appear curdled). Transfer batter to cake pans, even amount in each, and smooth tops. Bake until cake begins to pull away from side of pan and a wooden pick inserted into center comes out clean, 35 to 40 minutes. Cool cakes in pan 5 minutes, then invert onto a rack and cool completely, about 1 hour. Make Glaze FIRST, as it needs to cool for 30 minutes. For filling: 8 ounces, weight Cream Cheese, Softened 1 cup Powdered Sugar ⅛ teaspoons Salt 1 teaspoon Vanilla 1-½ cup Heavy Whipping Cream Make filling: Blend whipping cream on high in bowl until light and fluffy. Set aside. In another bowl, mix together on medium speed cream cheese, sugar, salt and vanilla. Fold in whipped cream. For glaze: 1/4 cup heavy cream 3 1/2 oz bittersweet chocolate finely chopped 2 teaspoons light corn syrup Make glaze: Bring cream to a simmer in a small heavy saucepan over medium heat, then pour over chocolate in a bowl and let stand 1 minute. Gently whisk until smooth, then stir in corn syrup. Cool completely, gently stirring occasionally, about 30 minutes (glaze will thicken). Put it all together: Put cake together by positioning one cake layer on a plate. Then spread on filling. Place second layer on top. Pour glaze over cake.
White Cake Calculus Cake #3B Ingredients: 2 1/4 cups cake flour (9 ounces), plus more for dusting the pans 1 cup whole milk, at room temperature 6 large egg whites (3/4 cup), at room temperature 2 teaspoons almond extract 1 teaspoon vanilla extract 1 3/4 cups granulated sugar (12 1/4 ounces) 4 teaspoons baking powder 1 teaspoon table salt 12 tablespoons unsalted butter (1 1/2 sticks), softened but still cool Preparation 1. Heat oven to 350 degrees. Spray two 9-inch round cake pans with nonstick cooking spray; line the bottoms with parchment or waxed paper rounds. Spray the paper rounds, dust the pans with flour, and invert pans and rap sharply to remove excess flour. 2. Pour milk, egg whites, and extracts into 2-cup glass measure, and mix with fork until blended. 3. Mix cake flour, sugar, baking powder, and salt in bowl of electric mixer at slow speed. Add butter; continue beating at slow speed until mixture resembles moist crumbs, with no powdery streaks remaining. 4. Add all but 1/2 cup of milk mixture to crumbs and beat at medium speed (or high speed if using handheld mixer) for 1 1/2 minutes. Add remaining 1/2 cup of milk mixture and beat 30 seconds more. Stop mixer and scrape sides of bowl. Return mixer to medium (or high) speed and beat 20 seconds longer. 5. Divide batter evenly between two prepared cake pans; using rubber spatula, spread batter to pan walls and smooth tops. 6. Bake until thin skewer or toothpick inserted in the center comes out clean, 23 to 25 minutes. 7. Let cakes rest in pans for 3 minutes. Loosen from sides of pans with a knife, if necessary, and invert onto wire racks. Let cool completely, about 1 1/2 hours. Frosting and Filling Ingredients: 2¼ cups powdered sugar ¼ cup unsweetened cocoa powder Pinch of Salt 6 oz cream cheese 1½ sticks unsalted butter, softened 9 oz bittersweet chocolate, melted and slightly cooled ¾ cup sour cream Preparation Sift together sugar, cocoa, and salt. Beat cream cheese and butter with mixer on medium-high speed until smooth. Reduce speed to medium low; gradually add sugarcocoa mixture, and beat until combined. Pour in chocolate in a slow steady stream. Add sour cream; beat until combined. Putting it all together: Place one layer of cake on serving dish. Cover the layer with half the frosting. Place the second layer on top. Cover the top with the remaining frosting.
13 tablespoons butter 1 1/8 cups sugar 3 eggs 2 1/4 cups flour 1 1/2 teaspoons baking powder 1/4 teaspoon salt 1 tablespoon +2 teaspoons orange juice concentrate Orange Cake - Calculus Cake #4A 2 tablespoons water 1/2 cup plus1 teaspoon milk 1/8 cup canola oil ¾ teaspoon vanilla extract Zest from one orange Directions: Preheat oven to 350 F. Spray two 9-inch round cake pans with nonstick cooking spray; line the bottoms with parchment or waxed paper rounds. Spray the paper rounds, dust the pans with flour, and invert pans and rap sharply to remove excess flour. Cream together butter and sugar until light and fluffy. Add eggs, one at a time, beating well after each addition. Sift together flour, baking powder and salt. Combine the orange juice concentrate, water, milk, canola oil, vanilla and orange extract. Add dry ingredients and wet ingredients, alternately, to butter mixture, beating well after each addition. Divide batter among cake pans and bake for 33 minutes or until toothpick inserted in center of cake comes out clean. Remove from oven, cool, and remove from pans. Frosting and Filling Ingredients: 1½ cups sugar 6 large eggs Pinch of Salt ¼ teaspoon cream of tarter 2 cups unsalted butter, cut into tablespoons 1 teaspoon pure vanilla extract Preparation: Heat sugar and eggs; Bring saucepan filled with about 2 inches of water to a simmer. Meanwhile, combine sugar, egg whites, and salt in a large bowl. Set bowl over (not in) simmering water, and whisk until whites are warm to the touch and sugar is dissolved, 2 to 3 minutes. Test by rubbing between your fingers; mixture should feel smooth. Whip meringue: Use electric mixer to beat on low speed until foamy (using whisk attachment). Add cream of tartar, and beat on medium-high speed until stiff, glossy peaks form and mixture is cooled completely. About 10 minutes. Mix in butter: Reduce speed to medium-low; add butter 2 tablespoons at a time, beating to incorporate fully after each addition. Don t worry if the buttercream appears curdled at this point; it will become perfectly smooth again with continued beating. After all the butter has been added, add the vanilla. Finish: Switch to a paddle attachment on the mixer and beat on lowest speed to reduce air bubbles, 3 to 5 minutes. Putting it all together: Place one layer of cake on serving dish. Cover the layer with half the meringue. Place the second layer on top. Cover the top with the remaining meringue.
Red Velvet Cake Calculus Cake #5E Ingredients: ¾ cup butter softened (12 tablespoons) 2 large eggs 1 teaspoon vanilla 1 teaspoon salt 2 1/2 cups all-purpose flour 1 1/2 cups sugar 2 tablespoons red gel food coloring plus 2 tablespoons water 2 tablespoons unsweetened cocoa 1 cup buttermilk 1 tablespoon vinegar 1 teaspoon baking soda. Preparation: Preheat oven to 350. Generously grease and line with parchment paper three 8-inch round layer cake pans or two 9-inch cake pans. Cream butter and sugar until light and fluffy. Add eggs and beat 1 minute longer. Beat in vanilla until blended. Combine cocoa, food coloring, and 2 tablespoons water; beat into the egg creamed mixture. Combine flour and salt. Alternate adding flour and buttermilk mixtures to the batter, beating on low to medium speed. Combine soda and vinegar in cup and add to cake batter. Spoon batter evenly into the 3 cake pans; bake at 350 for 22 to 28 minutes. Cool on cake racks. Frosting and Filling Ingredients: 4 Oz. Cream cheese 2 Teaspoons Milk 1½ Cups Whipping Cream ½ Cups Powdered Sugar Preparation: In chilled large bowl, mix cream cheese and milk until smooth. Beat in whipping cream and powdered sugar with electric mixer on high speed, scraping bowl occasionally, until soft peaks form. Putting it all together: Place one layer of cake on serving dish. Cover the layer with half the cream cheese frosting. Place the second layer on top. Cover the top with the remaining cream cheese frosting.