Question 1. Rate of changes in temperature
Record the daily temperature of Knoxville from August 1-August 5, 2015 and plot it in a graph with x-axis as date and y-axis as temperature. Connect the points by line segments. Using calculus, nd the average rate of change in temperature in the 11 days. Find the slope of secant line joining the end points of the plot. What is the unit of this slope? Compare this with the average rate of change in temperatures. Using calculus, determine whether the temperature of Knoxville is going up or down from August 3 to August 4. What is the rate of change of temperature at August 3. We do not know the equation of the graph. How do we approximate the slope of the curve at August 3? What is the sign of the slope and what does the sign mean?If f(x) is the equation of the plot, how would you nd the slope of f(x) on August 3 (instantaneous rate of change on August 3)?
Question 2. Implicit Di erentiation
Find the points on the graph of 3x2 + 3y2 + 3xy = 24 (both graphically and using calculus) where the tangent line is horizontal:
Graphically: Open the link https : ==www:wolframalpha:com and then type plot (3x2 + 3y2 + 3xy = 24). From the graph, nd points where the tangent is horizontal.
Using Calculus: Differentiate both sides with respect to x. Set y0 = 0 and get an equation in terms of x and y. Solve it for y and plug it into the given curve to get x. Plug in the value of x in the equation to get y value.
Question 3. Make a rectangle of greatest area
Take a rope of 4m length. Make rectangles of di erent dimensions(1.5,0.5), (1.2, 0.8), (1.1,0.9), (1,1), (0.5, 1.5), (0.8, 1.2(, (0.9,1.1)
Find the area of each rectangle. Which rectangle has greatest area?
Solve this problem using calculus to find the dimensions of a rectangle of given perimeter of 4m that includes maximum area.
Question 4. Construct a box of maximum volume from given square sheet of paper Take a square sheet of paper that is 8 8 inches. Make boxes by cutting squares (0.5, 1, 2, 3, 4, 5 inches) out of the corners and folding up the edges. Tape the edge together. Find the volume of all open boxes. Which box has the greatest volume?