9. (12 pts)Find the horizontal tangents to the graph of the cardioid r=1-cosθ, 0 ≤ θ ≤ 2π. (That is, find the points at which horizontal lines tangent to the curve.)

(a) (5%) ∫ dx 

(b) (5%) ∫ dx

(a) (10%) For a certain real number a, the following integral  converges. Determine a and evaluate the integral.

(b) (10%) For what values of the constants b and cwill the following limit exist and be equal to 1? dx .

3. (15%) If dis an arbitrary real number, let Sn(d) =. Determine the following limit: .

4. (10%) Let f(x) be a polynomial of degree m. Show that ∫exf(x)dx = ex(f(x) - f'(x) + f"(x) - ... + (-1)mf(m)(x))

5. (15%) Evaluate the following integral  along the curve C = {(x,y,z) ∈ R³ | x² + y² = 4, z = -3}, oriented counterclockwise as seen by a person standing at the origin and with respect to the right-handed Cartesian coordinates, where F = yi + xz³j – zy³k, r' = dr/ds is the unit tangent vector, and *sis the arc length of C

6. (15%) Evaluate the following integral I = ∫L(2xyz²) dx + (x²z² + z cos(yz)) dy + (2x²yz + y cos(yz)) dzalong the line segment L from P: (0,0,1) to Q: (2,π/4,2)

7. (15%) Compute the area of the hemisphere of radius t>0 and centered at the origin given by S = {(x,y,z) ∈ R³| x² + y² + z² = t², z≥0}

1.Source https://www.weforum.org/agenda/2022/02/future-of-the-metaverse-vr-ar-and-brain-computen/ In January 2020, author and investor Matthew Ball published the most commonly referenced description of the metaverse: "an expansive network of persistent, real-time rendered 3D worlds and simulations that [...] can be experienced synchronously by an effectively unlimited number of users, each with an individual sense of presence." According to his definition, the metaverse is a product or service, with seven core attributes including persistence, synchronicity, and interoperability. The technology strategist Ben Thompson has argued, however, that this description is in fact not far from what the internet already is and does, only "with a 3D layer on top of it". ............. The third definition is compelling in part because it focuses on those who will actually build and use the metaverse: people. Answering questions about how the metaverse might look and feel -rather than pondering about its features could prove a useful viewpoint to predict the wave of socioeconomic change that the metaverse is expected to unleash. After all, the future is built by ruthless pragmatists； not the armchair theorizers who "meander the forest of their own words".

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