Study with the several resources on Docsity
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Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
This is the Exam of Calculus which includes Constant, Constant Height, Concave Up, Antiderivative, Compare, Natural Domain, Average, Range, Natural Domain etc. Key important points are: Big Calculus, Graphs, Compute, Problem Combines, Fundamental Theorem, Calculus to Compute, Integral, Derivative, Limit, Demonstrate
Typology: Exams
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Your grade is based on correctness, completeness, and clarity on each exercise. Explain all answers completely. You may use a calculator, but no notes, books, or other students. Good luck!
1.) (10 pts.) Use the given graphs to compute L 3 and T 3. Simplify your answers.
a.) (5 pts.) On the graph below, sketch in and compute L 3.
1 2 3 4 5 6 x
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y
b.) (5 pts.) On the graph below, sketch in and compute T 3.
1 2 3 4 5 6 x
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y
2.) (15 pts.) This problem combines our three big calculus concepts: the integral, the derivative, and the limit.
a.) (5 pts.) Use the Fundamental Theorem of Calculus to compute
12 (x
− (^3) − 8) dx.
b.) (5 pts.) Compute the derivative of y = x−^3 − 8.
c.) (5 pts.) Compute the limit lim x→ 0 (x−^3 − 8). Be sure to demonstrate how you are com- puting the limit.
4.) (15 pts.) The following are two examples of techniques we have learned that require multiple steps.
a.) (7 pts.) Use L’Hˆopital’s Rule to compute lim x→ 01 −^ cos(3 8 x x). Be sure to confirm why you can use L’Hˆopital’s Rule.
c.) (8 pts.) Use implicit differentiation to compute dydx for the equation 2(x + y)^13 = y. Be sure to solve for dy dx
5.) (15 pts.) The function g(x) = 201 x^5 +^16 x^4 + 2 is shown in the graph below. You may choose to answer part (a) first, or wait and use your results from parts (b) and (c).
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gHxL
a.) (5 pts.) State the x-intervals for which the following hold. You can restrict g(x) to the x-values [− 3. 5 , 2], as shown in the graph.
i.) g(x) is increasing
ii.) g(x) is decreasing
iii.) g(x) is concave up
iv.) g(x) is concave down
b.) (5 pts.) Use calculus to show how you know the exact x-values at which stationary points occur.
c.) (5 pts.) Use calculus to find any inflection points and confirm that they are inflection points.
7.) (15 pts.) A rectangle initially has dimensions 2 cm by 4 cm. All sides begin increasing in length at a rate of 1 cm/s. At what rate is the area of the rectangle increasing after 20 s?
BONUS: (5 pts.) Many of you have already done this, but in case you have not yet: by 4:00pm today (Tuesday, April 10) email me a photo of yourself at the Mount David Summit.
