Calculus Answers

Consider the surface S = n (x, y, z) | z = p x 2 + y 2 and 1 ≤ z ≤ 3 o .(a) Sketch the surface S in R 3 . Also show its XY-projection on your sketch. (2) (b) Evaluate the area of S, using a surface integral

Question No#08

Find the limit of each rational function as 𝒙 → ±∞. 𝑓(𝑥) = 3𝑥 + 7

 𝑥2 − 2

𝑓(𝑥)= 10𝑥5 +𝑥4 +31. 𝑥6

Question No#10

Let

ƒ(x) = 𝑥1/(1−x) .

Make tables of values of ƒ at values of x that approach x = 1 from above and below. Does ƒ(x)

appear to have a limit as x approaches 1? If so, what is it? If not, why not?

 Question No#11

Find the limits using 𝐥𝐢𝐦 𝐬𝐢𝐧 𝜽 = 𝟏 𝒙→𝟎 𝜽

a) lim 6𝑥2(cot 𝑥)(csc 2𝑥) 𝑥→0

𝑏) lim sin 3𝑦 cot 5𝑦 𝑦→0 𝑦 cot4𝑦

Question No#13 Let h(𝑥) = 𝑥2−2𝑥−3

a) Make a table of the values of h(x) at x = 2.9, 2.99, 2.999, and so on. Then estimate limh(𝑥). What estimate do you arrive at if you evaluate h at x = 3.1, 3.01, 3.001 ...

𝑥→3

instead?

b) Support your conclusions in part (a) by graphing h near x = 3 and using that graph to

estimate h(x) on the graph as x approaching 3.

c) Find lim h(𝑥) algebraically.

𝑥→3

Let 𝑔(𝜃) = sin 𝜃 𝜃

a) Make a table of the values of g at values of u that approach 𝜃0 = 0 from above and below. Then estimatelim 𝑔(𝜃)

𝜃→0

b) Support your conclusion in part (a) by graphing g near 𝜃0 = 0

a) Graph 𝑔(𝑥) = 𝑥 𝑠𝑖𝑛(1⁄𝑥) to estimate lim 𝑔(𝑥), zooming in on the origin as necessary 𝑥→0

(b) Confirm your estimate in part (a) with a proof.

Suppose that the speed v (in ft/s) of a skydiver t seconds after leaping from a plane is given by the equation 𝑣 = 190(1 − 𝑒 − 0.168𝑡).

(a) Graph 𝑣 versus 𝑡.

(b) By evaluating an appropriate limit, show that the graph of 𝑣 versus 𝑡 has a horizontal asymptote 𝑣 = 𝑐 for an appropriate constant 𝑐.

(c) What is the physical significance of the constant 𝑐 in part (b)?

Let

ƒ(x) = 𝑥1/(1−x) .

Make tables of values of ƒ at values of x that approach x = 1 from above and below. Does ƒ(x)

appear to have a limit as x approaches 1? If so, what is it? If not, why not?