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Earn points by helping other students or get them with a premium plan
This is the Exam of Calculus which includes Find, Differentiable, Function, Limit Definition, Derivative, Limits, Evaluate, Calculus, Respect, Elliptic Track etc. Key important points are: Extreme Value Theorem, Function, Interval, Geometry, Calculus, Compute, Cylindrical, Minimize, Dimensions, Manufacture
Typology: Exams
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Your grade is based on the process as well as the final result. Show all your steps clearly so you will be eligible for the most partial credit. You may use a calculator, but no notes, books, or other students. Good luck!
1.) (10 pts.)
a.) (5 pts.) What does the Extreme Value Theorem say about the function f (x) =
9 โ x^2 on the interval [0, 3]?
b.) (5 pts.)โซ Use either geometry or the Fundamental Theorem of Calculus to compute 3 0
9 โ x^2 dx.
2.) (15 pts.) A cylindrical can (with top) is to be made to hold 1 liter of oil. Find the dimensions that will minimize the cost of the metal to manufacture the can.
4.) (15 pts.)
a.) (5 pts.) State the hypotheses of the Mean Value Theorem.
b.) (5 pts.) State the conclusion of the Mean Value Theorem.
c.) (5 pts.) Sketch a function and appropriate interval that satisfy the Mean Value The- orem. Be sure to show that both sides of the equation in the MVTโs conclusion are equal in your sketch.
5.) (15 pts.) The equation y^2 = x^3 (2 โ x) has the shape of a piriform, shown in the graph at the bottom of this page.
a.) (5 pts.) Compute
dy dx
for the piriform equation.
b.) (5 pts.) Find the equation of the tangent line to the piriform at the point (1, 1).
c.) (5 pts.) Sketch the tangent line at (1, 1) on the graph below. Does it agree with your result in part (b.)? Explain.
โ0.
0
1 y
0.5 1 1.5 2 x
7.) (15 pts.) Draw a single function f (x) in which all of the following are true:
i.) lim xโโ 3 โ^
f (x) = โ 1
ii.) lim xโโ 3 +^
f (x) = 2
iii.) lim xโ 1 f (x) = โ
iv.) lim xโโ f (x) = โ 5
BONUS: (5 pts.) Which high school dropout was able to teach us something about Calculus AND what did we learn from him?
