Evaluate the integrals using appropriate substitutions

Question

Question 1. Find the limits. Give the exact answer.

a.     lim    ln (sin x)/ ln ( tan x)
    x →0⁺

b.   lim     (1 - 2/x )^x
    x →+∞

c.   lim       sec5xcos9x =
    x →Π/2^-

Question 2. A positive number ∈ and the limit L of a function f at a are given. Find a number δ such that |f (x) -L| <  ∈ if 0 < |x - al < δ.

   lim    x³ = 8; ∈ = 0.001
   x →2

Question 3. Find values of x, if any, at which f is not continuous.

f (x) =   2x + 3, x ≤ 4
            7 + 16/x, x>4
Show full reasoning as to why.

Question 4. For this problem find the following characteristics of the function and then graph the function.

a. Finding critical points

b. Naming relative maximum(s) and minimum(s), then naming the absolute max and absolute min if they exist.

c. Indicating intervals of increasing and decreasing

d. Indicating intervals of concave up or concave down

e. Indicating inflection points

f. Graphing the function with above characteristics and any asymptotes/end behavior.

sin x + cos x  on interval x: [-2pi, 2pi]

Question 5.

A cylindrical can, open at the top, is to hold 500 cm³ of liquid. Find the height and radius that minimin the amount of material needed to manufacture the can.

Volume = pi * r^2 * h

Surface Area = 2*pi*r^2 + 2*r*pi*h

Question 6. Find d²y/dx² by implicit differentiation.

y + siny = x

Question 7. A 13 ft ladder against a wall. If the top of the ladder slips down the wall at a rate of 2 ft/s, how fast will the foot be moving away from the wall, when the top is 5 ft above the ground?

Question 8. Evaluate the integrals using appropriate substitutions.

a. ∫cos⁴3t sin 3t dt

For full credit double checking by taking the derivative is necessary.

b. ∫ dx/√xe^(2√x)

For full credit double checking by taking the derivative is necessary.

c. ∫ (e^x/ 1 + e^2x).dx

For full credit double checking by taking the derivative is necessary.

d. ₄∫⁹ 2x√xdx

For full credit double checking by taking the derivative is necessary.

e. ₀∫^Π/2 4 sin(x/2)dx

For full credit double checking by taking the derivative is necessary.

f. _-1∫¹ x²dx/(√x³ +9)

For full credit double checking by taking the derivative is necessary.

Question 9. Use a graphing utility, where helpful, to find the area of the region enclosed by the curves.

a. y = sinx, y = cosx, x = 0, x = 2Π

b. x = y³ - 4y² + 3y, x = y² - y