Calc I 13 Questions

Mathematics

### Question Description

There are 10 regular 3 extra credit questions. I need them all be completed and also I need the solutions with the answers on a paper (hand-written) and you can send me that paper in pdf format. It has to be pdf format.

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Name (print): Answers go on the test pages. Show all work  on a separate sheet if necessary. Answers given without your own supporting work may not receive credit. Please circle your answers. Good luck! 1. (15 pts.) Evaluate the limit, if it exists (for innite limits, write (a) (b) (c) (d) (e) ∞ or −∞): lim (x17 − x + 3) x→1 lim x→2 x2 +x−6 x−2 lim 6 x−5 lim √ 5x 2x2 +1 x→5− x→∞ lim h→0 (5+h)2 −52 (Extra Credit if you can identify h 2. (8 pts.) Find the what function this is the derivative of and at what value.) derivative of the following functions: 3 sin x x . (a) f (x) = (b) y = sin(cos(x)) 3. (8 pts.) Find an equation of the 4. (8 pts.) Use tangent line to the curve y = x 1 + x2 at the point √ (1, 2). implicit dierentiation to nd y0 in terms of x and y for the following equation: x2 y3 +3y2 = x−8y. 5. (10 pts.) A balloon is rising at a constant speed of 15 √ 5 ft/s. A boy is cycling along a straight road at a speed of ft/s. When he passes under the balloon, it is 45 ft above him. How fast is the distance between the boy and the balloon increasing 3 seconds later. 6. (15 pts.) For the curve (a) Calculate y0 and y = 10x4 − x5 : y 00 . (b) Use the rst and/or second derivative test to nd the x values of all local extrema (mins and maxes). (c) Find all points of inection and indicate them on a sketch of y. (d) Find the (absolute) minimum and maximum values of f (x) on the interval [0, 2]. 7. (10 pts.) Consider the function f (x) = 15x2 + 2x on the interval [0, 1]. (a) Approximate the integral of the function over the interval by using the with 5 (b) Approximate the integral of the function over the interval by using the with 5 left endpoint approximation rectangles (L5 ). right endpoint approximation rectangles (R5 ). (c) Evaluate the integral exactly and compare with you answer in parts (a) and (b). R5 R5 g(x) dx = 5, 7 g(x) dx = 2π , and 3 g(x) dx = 3 + 4π , might help to think of the integrals as areas.) 8. (6 pts.) If 9. (20 pts.) R3 1 nd R7 1 g(x) dx. (Show work.) (Hint: It Use the Fundamental Theorem of Calculus to evaluate the following (answers must be exact, not decimal approximations): (a) (b) (c) (d) R5 3 R9 1 ex dx 1 t− 2 dt R π/3 0 R −1 −2 sin t dt x−1 dx 10. (Extra Credit) Answer any of the following: (a) What nefarious name is given to the curve y = 1/(1 + x2 ) ? (b) What product was introduced by Francis Ford Coppola's company in 2017 that honors the 18th century mathematician who studied this curve (and developed the derivation of the quotient rule we used in class)? (c) Who invented the S-shaped integral sign ( R ) and what does the S stand for? 11. (Extra Credit) Find the area of the largest rectangle that can be inscribed in a 12. (Extra Credit) Suppose f (t) semicircle of radius represents the total number of conrmed cases of a virus at time t √ 2. (in days), and f (0) = 10, 000. (a) How many days would it take for constant at 30%? 3%? f (t) 1, 000, 000 cases if the relative 0 of growth (f (t)) at this point? rate of growth (f 0 (t)/f (t)) is 1, 000, 000 cases if the relative 0 (t)) at this point? rate of growth (f 0 (t)/f (t)) is to reach What is the absolute rate (b) How many days would it take for constant at f (t) to reach What is the absolute rate of growth (f 13. (Priceless) Have a good summer! ...
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