Old Final Exam

1. [9 pts] Find the derivatives of the following:
a) y = (2x + 3)²(5 – 3x)
b) y = 3x – 4
          4x + 8
c) y = sin(3x²)

2. [5 pts] Find the limit (if it exists).

lim          x – 2
^{x ->2}   x² + 2x – 8

3. [5 pts] Find the limit (if it exists).

       lim          x – 2
^{x ->∞}      x² + 2x – 8

4.[5 pts] Find the equation of the tangent line to the function f(x) = 5x² - 3 at the point

(1, 2).

5.  [10 pts] Find dy/dx by implicit differentiation.
          cos(x) + xy² = 3x³

6. [9 pts] Given

                      f(x) = 5         if x = 1
         3 + x     if x < 1
        x² + 3     if x > 1

a. Sketch the graph of f(x)

b. Find          lim f(x) and lim f(x)
^{x ->1 -                x -> 1+}

c.  Does lim f(x) exist? Is f(x) continuous at x = 1? Why or Why not?
              ^{x ->1}

7. [9 pts] If a snowball is melting at a rate of dA/dt = 2cm²/min, find the rate at which the radius decreases (dr/dt = ?) when the diameter is 5 cm?

(Hint: The formula for surface area of a snowball is A = 4π r²)

8. [10 pts] Find all local maxima and minima and absolute maxima and minima of f(x) = 3x² – 12x + 5 on the interval [0, 5].

9. [10 pts] Evaluate.

lim       2x – sin x
^x-->0     cos(x) - e^x

10.[6 pts] Find the derivative of y.

a) y = _x ∫ ^-2 (2tan x - 4)dx

b) y = _-3 ∫^3x (3x – ln x)dx

11. [12 pts] Evaluate the following integrals:

a. ∫ sec²x dx =

b. ∫ 1 – x – 3x² dx
             x²

c. ∫ (x - 4)² dx =

d. ∫ 2 dx =

12. [10 pts] Evaluate the following indefinite integrals using substitution.

a. ∫ (2x - 4)⁴ dx =

c. ∫ e^{cos 2x} sin2x dx =