Calculus Answers

Unit 1 Review:

7) Use link below to answer questions

• Even or odd degree?
• Least possible degree?
• Number of turning points?
• Positive or negative leading coefficient?
• # of real zeros?
• Coordinate y intercept

Unit 1 Review:

5) What is domain of

f(x) = { 3x + 2, -3 < x < 1

-2x + 5, 1 < x < 4

6) What is range of

f(x) = { 3x+2, -3 < x < 1

-2x + 5, 1 < x < 4

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?

Use the Rational Zeros Theorem to write a list of all possible rational zeros of the function.

f(x) = -2x⁴ + 4x³ + 3x² + 18

Show that |∫_−2𝜋^2𝜋 𝑥 ² sin⁸ (𝑒 ^𝑥 ) 𝑑𝑥| ≤ (16𝜋 ³ )/3

determine the maximum profit in the price that would yield the maximum profit for P=-400p^2+12,400p-50,000

The distance between cities X and Y is equal to 21 kilometers. A pedestrian leaves X and goes to Y at a constant speed of 5 kilometers per hour. At the same moment, a cyclist leaves Y and goes towards X, the speed of the latter can vary between 10 and 13 km/h throughout the journey. After meeting each other, the motorcyclist goes 26 minutes more towards city X, and then turns back and returns to Y. What is the maximum difference in time the pedestrian and the cyclist arrive to Y? Express the answer in minutes.

A new pizzeria has opened near you home, the owner has decided to offer a 20% discount on

the opening day on the pizza whose original price is x dollars. The owner is offering additional

15% discount coupon to those who agree to fill a feedback form afterwards. You and your

friends have agreed to provide feedback and thus have become entitled to the additional 15%

discount coupon. Use the concept of composite functions to identify the amount that you will

pay after applying the coupon on the sale price.

a) Give parametric equation (point-direction form) of the line which lies on both of the planes:

x + y + z = 1 and −x + 2y + 10z = 2. What is the direction d of this line?

b) Let n1 and n2 be the normal vectors to the two given planes. Without actual computation,

describe the relationship between d and n1 × n2.