Calculus Answers

let

f(x) =1 + 2𝑥, 𝑥 ≤ 0

=3𝑥 − 2,0 < 𝑥 ≤ 1

= 2𝑥 ଶ − 1, 𝑥 > 1

i) Check whether f is discontinuous. If yes, find where? ii) Give a rough sketch of the graph of f. 

Find the second derivative of 3y^4+x^7=5x

Determine the length of the parametric curve given by the following parametric equations. x=3sin(t) y=3cos(t) 0<t<2π

Evaluate the following line integrals along with the curve C.

(a) ∫

C (x² − 2y² ) ds; C is the line segment parametrized by r(t) = < t / √ 2 , t / √ 2 >, for 0 ≤ t ≤ 4.

Evaluate the following line integrals along with the curve C.

(a) ∫

C (x² − 2y² ) ds; C is the line segment parametrized by r(t) = < t √ 2 , t √ 2> for 0 ≤ t ≤ 4.

Determine whether the following vector fields are conservative on R 2 . If the vector field is conservative then find a potential function. F =<e^x cos y, −e^x sin y>.

Trace the curve y²=(x-a)(x-b)(x-c) with a, b, and c are all positive.  

a²x²=y³(2a-y) trace the curve

following statement is true and which are false? Give reasons for your answers, in the form of a short proof or a counter example.

Every continuous function is differentiable

DM. Let a and b be two cardinal numbers. Modify Cantor’s definition of a < b to define a ≤ b. (Hint: Examine what happens if you drop condition (a) from Cantor’s definition of a < b.) 2. Prove that a ≤ a. 3. Prove that if a ≤ b and b ≤ c, then a ≤ c. 4. Do you think that a ≤ b and b ≤ a imply

a = b? Explain your reasoning. (Hint: This is not as trivial as it might look.)