Math calculus

Math 5A, calculus, Mrs. Badalyan FINAL EXAM 6/8/20, 7:45-9:45 pm. ONLINE Print Name_________________

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Show your very detailed work. Do not skip any part of the problem. Show all justifications of your

conclusions and answers. Be as accurate as possible not to miss any points. Please be advised it is

limited to two hours only.

1.(10pts) Find the numbers at which function f is continuous if any, at which it is continues only from

the right, from the left, or neither. Clearly graph the function in the space provided. Justify your answers

by checking all three criteria for continuity. Make your conclusion based on those three criteria clear and

underlined. 𝑓 =

2𝑥2−7𝑥+3

𝑥−3 , − 2 ≤ 𝑥 < 3

6 − (𝑥 − 4)2, 3 ≤ 𝑥 < 6 −2𝑥 + 14, 𝑥 > 6

2.(5 pts) Find the limit lim 𝑥→0

𝑥

1−√1+3𝑥

3(5 pts).Find 𝑑𝑦

𝑑𝑥 𝑏𝑦 𝑖𝑚𝑝𝑙𝑖𝑐𝑖𝑡 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑡𝑖𝑜𝑛:

tan(𝑥 − 𝑦) = 𝑦

1+𝑥2

Math 5A, calculus, Mrs. Badalyan FINAL EXAM 6/8/20, 7:45-9:45 pm. ONLINE Print Name_________________

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4. (5 pts) A particle is moving along a hyperbola xy=8. As it reaches the point (4,2), the y-coordinate is

decreasing at the rate of 3 cm/sec. How fast is the x-coordinate of the point changing at that instance?

Sketch the graph.

5. (4+1 pts) 5a.Find the derivative of 𝑦 = cos2( 3𝑥 − 5)2. Simplify as much as possible.

5b. write the equation of the tangent line to the point (5/3,1).

6. (5 pts) Find the point on the curve 𝑦 = √𝑥, 𝑡ℎ𝑎𝑡 𝑖𝑠 𝑐𝑙𝑜𝑠𝑒𝑠𝑡 𝑡𝑜 𝑡ℎ𝑒 𝑝𝑜𝑖𝑛𝑡 (3,0) Sketch the graph, Verify

that you found the extreme point.

7. (5 pts) Find the derivative of the function 𝑦 = ∫ cos𝑥

3𝑥+1 √𝑡 + sin𝑡𝑑𝑡

8. (5 pts) Set up and do not evaluate. Find the volume of the solid obtained by rotating the region

bounded by the given curves about the specified line. Sketch the region, by

finding x and y intercepts, HA and VA asymptotes (no limits are required). Show

the slicing, indicate the method you are using. 𝑦 = 𝑥, 𝑦 = 2𝑥

1+𝑥3 𝑎𝑏𝑜𝑢𝑡 𝑥 = −1

Math 5A, calculus, Mrs. Badalyan FINAL EXAM 6/8/20, 7:45-9:45 pm. ONLINE Print Name_________________

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9. (5pts) Evaluate. Use the washer method to find the volume generated by rotating the region bounded

by the given curves about the specified axis. 𝑦 = √𝑥, 𝑥 = 2𝑦, 𝑎𝑏𝑜𝑢𝑡 𝑥 = −3

10. (10 pts) Graph the function 𝑦 = 1 + 1

𝑥 +

1

𝑥2 Clearly indicate domain, VA (use the

limits),HA( use the limits), first derivative, state the Intervals of

increase decrease, extremes; second derivative, intervals of CU/CD,

IP. Provide a detailed graph.