Calculus Answers

Provide all detailed steps to find the limit of the following functions.

Lim x→∞ (e^4x - e^-2x) ÷ (ln(x+1)

F(x) = 2x³ + cx² + 2x.

Suppose f is differentiable on R and has two roots. Show that f' has at least one root.

1. The demand for a product, in dollars, is P = 2000 - 0.2x - 0.01x²

Find the consumer surplus when the sales level is 250.

2.1. Find the area of the region that lies inside the circle r = 3sin(x)and

outside the cardioid r = 1+ sin(x).

2.2 Find the length of the cardioid r = 1+ sin(x)

1.Find the Maclaurin series for thefunction f(x)= (x^2+ 2-x)^5/2and its radiusofconvergence

f(x,y)= -(x²-1)²-(x²y-x-1)²

Find the critical points of f (x, y) and Using the Second Partials Test (SPT), find the relative extrema’s of f (x ,y)

Q.4) If an object starts out at rest and falls a distance of s feet in t seconds, then physical laws imply that s = 16t². Find ∆s/∆t as t changes from t0 to t0 + ∆t.

Use the result to find ∆s/∆t as t changes from:

(a) 3 to 3.5, (b) 3 to 3.2, and (c) 3 to 3.1

find the derivative using chain rule.

1. y= (x³ + 3)⁵

2. y=(-3x⁵ + 1)³

3. y=(-5x³ - 3)³

4. y=(5x² + 3)⁴

5. f(x)=⁴√-3x⁴-2

6. f(x)= √-2x² + 1

Suppose that a particle moves along a line with position function s(t) =2t^2 +3t+1 where s is in meters and t is in seconds.

a. What is its initial position?

b. Where is it located after t = 2 seconds?

c. At what time is the particle at position s = 6?

For what values of 𝑐 does the curve 𝑓(𝑥) = 2𝑥3 + 𝑐𝑥2 + 2𝑥 have maximum and minimum

points?