Calculus Answers

Let D = {(x, y) ∈ R²: x > 0, 0 < y < x³}. Define

f(x, y) = {0, (x, y) ∈/ D

{1, (x, y) ∈ D.

a) Approaching (0, 0) along with the line y = mx for each real number m and the y-axis, prove that lim(x,y)→(0,0) f(x, y) exists and compute the limit.

b) Argue whether f is continuous at (0, 0).

Can the surface x⁵y³ + x²y + z = 3 be expressed as the graph of a function x = f(y, z) near the point (2, 1/2, −3)? If yes, compute f_y(1/2, −3) and f_z(1/2, −3).

a) Find the linear and quadratic approximations of sin(0.96π) tan(0.26π) + (0.96)²(0.26)

b) Compare these approximations with the actual value.

Let f(x, y, z) be a differentiable function. At the point (1, 1, 2), the directional derivative is 4,3,2 in the direction i + j, j + k and i + k, respectively.

a) Find the directional derivative at the point (1, 1, 2) in the direction 3i + 3j + 3k.

b) Compute ∇f(1, 1, 2).

c) In which direction does the function f increase most rapidly? In which direction does the

function f decreases most rapidly?

A company that operates for 18 hours a day has introduced a new daily wage system that incorporates variable hourly rate to encourage longer shifts. The company pays $10 for first hour or any part of the first hour. Afterwards, company pays additional 2% per hour for each hour or part of each additional hour. Write a piece-wise function that shows the salary as a function of hours.

Question 4

A vehicle has a 20-gal tank and gets 15mpg. The number of miles N that can be driven depends on the amount of gas 𝑥 in the tank:

a) Write a formula that models this situation.

b) Determine the number of miles the vehicle can travel on (i) a full tank of gas, and (ii)

3/4 of a tank of gas.

c) Determine the domain and range of the function.

d) Determine how many times the driver had to stop for gas if she has driven a total of

578 mile

Consider the function y=\ ln(1+x).

Which of the following is the Maclaurin series expansion of the first three terms?

1. "x-\\frac{x^2}{2}+\\frac{x^3}{6}"
2. "x-\\frac{x^2}{3}+\\frac{x^3}{2}"
3. "x-\\frac{x^2}{2}+\\frac{x^3}{3}\u200b"
4. "x+\\frac{x^2}{2}-\\frac{x^3}{3}"

Consider the following equation "I=\\displaystyle{\\int_{0}^{1}\\sin{(x^2)}dx}"

Find the integral  "\\displaystyle{\\int_{-1}^{1}\\sin{(x^2)}dx}" in terms of "I"

True or false:

The binomial coefficient "\\displaystyle{{-1\/2}\\choose{5}}" is equal to "\\dfrac{63}{256}"

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