Integration-Related-Rates-Optimization-Curve-Sketching - Engineering Assignment Help

Assignment Task :

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Question 1 - Curve sketching

2. Determine the intervals on which the function is increasing and decreasing. Identify all critical points and classify them as local minima or maxima if possible. Justify your answer.

3. Determine the intervals on which the function is concave up and concave down. Identify any points of in ection. Justify your answer.

•  Sketch a graph of the function including all of the information above.

Question 2 - Implicit Di erentiation

[2] Find the coordinates of all point(s) on the curve that have a horizontal tangent line.
[3] Find the coordinates of all point(s) on the curve that have a vertical tangent line.

 

Question 3 - Related Rates
[1] A cylindrical length of wire has a radius of 4 mm and a length of 10 cm. If the length is growing at a rate of 2 cm/sec and the radius is shrinking at a rate of 1 mm/sec, what is the rate of change of the volume in cm3/sec at that point in time. (Be careful of units)

[2] Consider the same length of wire as before (radius of 4 mm and length of 10 cm). This time we are stretching the wire at a rate of 2 cm/sec. If the total volume is not changing, what is the rate of change of the radius in mm/sec at this point in time.

Question 4 - Optimization
[1] The cost of fuel to propel a boat through the water (in dollars per hour) is proportional to the cube of the speed. A certain ferry boat uses $100 worth of fuel per hour when cruising at 10 miles per hour. Apart from fuel, the cost of running this ferry (labor, maintenance, and so on) is $700 per hour. At what speed should it travel so as to minimize the cost per mile traveled?

Question 5 - Limits

a) [1] The graph of f and g are shown together with their tangents lines at (1; 0). Compute

b) [2] For what value(s) of k does the following limit exist? Compute the limit for those value(s).

Question 6 - Derivatives
[5] Use Logarithmic Dierentiation to compute the derivative of the following function. Your nal answer can contain f(x) if you like. You do not have to simplify.

Question 7 - Function Families
[6] Suppose that a potato is immersed in a large pot of boiling water. After 10 minutes the potato has an internal temperature of 70C. After 20 minutes the potato has an internal temperature of 80C. The longer the potato stays in the water, the closer its internal temperature gets to to 100C. Model the internal temperature of the potato (in C) as a function of time spent in the water (measured in minutes) using a function of the form

[1] What was the internal temperature of the potato when it just entered the water?

Question 8 - Integration
[1] Suppose that the speed of a runner is measured every 100 meters along an 800 meter long track. Assuming that the speed was never decreasing, nd the best possible upper and lower estimate for the time (in seconds) it took the runner to run across the length of the track.
 

[2] How often would the speed need to be measured in order to estimate the duration of the run to within 1 second?

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