Calculus

2) Maple syrup is being pumped into a cone-shaped vat in a factory at a rate of six cubic feet per minute. The cone has a radius of 20 feet and a height of 30 feet. How fast is the maple syrup level increasing when the syrup is 5 feet deep?

(6) Olivia is vacationing in a cabin along a river. The path of the river near Olivia's cabin can be modeled by the function y = r². Olivia's cabin is located at the point (0,3) on the diagram. He noticed that a boat is floating at the point (3,2.25) on the river. River 2 Cabin (a) What is the distance of the boat from Olivia's cabin? (3.2.25) (b) What is the minimum distance from Olivia's cabin to the river? (c) What are the location(s) on the river that are closest to Olivia's Cabin? Page 3 of 4

For two rectangles, area For four rectangles, area Question 11 of 21 (Type an integer or a decimal.) (Type an integer or a decimal.) This test: 21 point(s) possible This question: 1 point(s) possible Using rectangles whose height is given by the value of the function at the midpoint of the rectangle's base (the midpoint rule), estimate the area under the graph of the following function, using two and then four rectangles. y = 9-x² between x = -3 and x = 3 Submit test

Question 2, 6.1.15 Part 1 of 4 HW Score: 16.59%, 7.13 of 43 points Points: 4.2 of 7 Save m S Consider an object moving along a line with the given velocity v. Assume t is time measured in seconds and velocities have units of Complete parts a through c. a. Determine when the motion is in the positive direction and when it is in the negative direction. b. Find the displacement over the given interval. c. Find the distance traveled over the given interval. v(t)=2t² - 16t+24; [0,7] a. When is the motion in the positive direction? Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. O A. For t-values that satisfy (Use a comma to separate answers as needed. Type your answers in interval notation.) O B. The motion is never in the positive direction.

¸¡₁(g(x))• g'(x) dx = p) Use the Substitution Formula, f(g(x)).g'(x) dx = a g(a) 2 In x X 2 Inx -dx dx = (Type an exact answer.) f(u) du where g(x)=u, to evaluate the following integral. C...

y" - 4y + 4y = 0; y(0) = 2,y' (0) = 1 Use Laplace Transformation to solve it. Do not use any other method. show all steps, copy the part of Laplace table that you are using.

Verify § ².d ²³ = SS (²x²). 1³² с that is show that the LHS =RHS, for in the surface The heat Z ph I deform, Stokes Theoran Ĵ 270 S: x² + y² +2²=3 =>0 trick that I use 골 10~ S31 7²x TX 7²x Imagine a bubble on a CIRCULAR rim C₁, Here &c, closed curve = §₁²₂² = § ²3₂ = § ₁₂₁ For a vector fuld we will find different surface SS (0x²) ds₁ = ²d² | C₁=C₁=C3=C4/ & 215 1₁² That for each the bubble so that 11 =12² in (14) only x Si 6₂ (0x²) Sf₂ (0x²).ds₂ = Fid²³, SS (oxf) ds = ) & F.d have to calculate the area of a circle!/nSo by luoking at my diagrams you can appreciate for each unusual Irregular smooth /bubble surface S₁, S₂ and 53 a) ff (√x²).d³₁ = √ ².dr SI C₁ 1) $ (0ײ).d³₂ = § ³.4² = & Fd² S2 2 =) ff ( x ²).d.² = & F₁d² = §£ F.d² CI S3 VX² I have F.d d d) $ (x²)² = & fid² = 6.² Ju JC4 a) = b) = c) = d) S4 but in of F. d² the bubble has been flattened planar surface and 1=1 here ₁ to a So f 11 where EASIEST TO SOLVE Alix да C4 î = F₁T +2₂3 +5h => (PXF) onds = (F₁, F₂, F) 12 => SS (Gx²³²) •nds = SS F3 ds F3 ds == ²0 and integral is easier. X

3. Find the smallest number M so that f(3) (c)| ≤ M for 8 ≤ c ≤ 10.

9 Find the following indefinite or definite integrals. (a) (4 points) (3r5 −1+) dx I (b) (4 points) f/4 sec²(x) dx (c) (6 points) Find the indefinite integral using the substitution method. [(3x²- +2) cos(x³ + 2x + 1) dr

(6a) (4 points) Differentiate the function f(r) = ³ sec(r). (6b) (4 points) Differentiate the function f(x) = 2x²+1 (6c) (4 points) Differentiate the function g(x) = [e+tan(z)]7. (6d) (4 points) Differentiate the function (x) = ln(x²+x+10).