Reference no: EM13902086

Question 1- On the region 0 ≤ θ ≤ 2π, find and classify all critical points and then identify the region where each function is concave up for each of the following.

(a) sin (x)

(b) cos (x)

(c) x³

Question 2- Using the limit definition of the derivative, show the following are true:

(a) If f(x) = 5, then f'(x) = 0

(b) If f(x) = x², then f'(x) = 2x

Question 3- Explain what it means for a function to be continuous at a point in terms of limits.

Question 4- What is the 2015^th derivative of f(x) en f(x) = cos (x).

Question 5- Let sinh(x) = (e^x - e^-x/2) and cosh(x) = (e^x + e^-x/2). Finf the first and second derivatives of sinh(x) and explain, briefly, how the pattern of derivatives of sinh(x) you found here relates to the pattern of derivatives you know about sin(x).

Question 6- Find a value of x that satisfies x + 2 = e^x, if one exist, using Newton's Method for finding roots.

Question 7- Using local linearizion, estimate the value π² if π ≈ 3.1415.

Question 8- Han Solo is piloting his spaceship, the Millennium Falcon, down a trench of the Death Star while Darth Vader is piloting his spaceship, a TIE Fighter, down a perpendicular trench. Both ships are travelling towards the same intersection. Suppose at a given moment, Han Solo is travelling at 60 imperial units per minute and is 8 units away from the intersection and Darth Vader is traveling at 30 imperial units per minute and is 6 units away from the intersection. Calculate how fast the ships are approaching each other at that moment.

Question 9- Using u-substitution, find the anti-derivation of x · e^x2.

Question 10- Evaluate the integrals:

(a) _-1∫¹x² dx    

(b) _-10∫¹⁰ sin(x) dx

(c) ₀∫⁴ e^2x dx

Question 11- Let f(x) = ₀∫^x2 sin(cos(t)) dt. Evaluate f'(√(π/2)).

Question 12- Consider the function f(x) = ₀∫^x e-(t+1)² dt and the fact that f(0) = 0. Use local linearizion to estimate the value of f(0.2).