1. [5 pts] Find the limit if it exists.

lim e^{cos x}

^{x ->0}

2. [9 pts] The table below gives the number of shopping centers, N(t), in the U.S. for selected years. Let t be the number of years since 1980.

a) Find the slope of the secant line from 1991 to 1992.

b) Find the slope of the secant line from 1992 to 1993.

c) Estimate and interpret N '(1992).

3. [10 pts] Find the equation of the tangent line to the function y = 3x² - 2 at the point (1, 1).

4. [10 pts] Find the limit (if it exists).

lim  (x² – 1)/ (x² + 2x - 3)

^{x ->1}

5.[10 pts] Find the limit (if it exists).

lim     sqrt(x + 2) - sqrt(2x)
^{x ->2}            2 – x

6. [5 pts] Use the squeeze theorem to prove that lim x³ sin(2/x) = 0.
                                                                         x -> 0

7. [12 pts] Determine the average rate of change of the function

f(x) = 2 - 3x²  between

a. x = 1 and x = 2
b. x = 1 and x = 1.1
c. x = 1 and x = 1.01
d. Estimate the instantaneous rate of change of f(x) at x = 1

8. [15 pts] Given

f(x) = x – 2     if x > 2

          4 – x²       if x < 2

a. Sketch the graph of f(x)

b. Find lim f(x) and lim f(x)

           ^{x ->2 -               x -> 2+}

c. Does lim f(x) exist? Is f(x) continuous at x = 2? Why or Why not?

            ^{x ->2}

9. [10 pts] Find the limit if it exists.

lim   (3x² – 1)/(4x² – 2x - 3)

^{x -> ∞} 

10. [10 pts] Given

f(x) =  (3x – 1)/(x - 3)

a. Find the vertical asymptote(s) (if any exist)

b. Find the horizontal asymptote (if it exists).

11. [4 pts] Use the limit definition to find f '(1) for the function f(x) = 3 – 4x.

Test 2

1. [9 pts] Sketch the graph of f(x) = (x – 4)(1 + x)(2 – x), -10 < x < 10 and directly below it sketch a rough graph of its derivative f '(x). (You may use your calculator to get the graph for f(x).)

2.[9 pts] For what values of x does the y = 2x³ + 3x² – 12x + 1 have a horizontal tangent?

3. [10 pts] Find dy/dx by implicit differentiation.

4cos x sin y = 1

4. [10 pts] Find the first, second, and third derivatives of f(x).

f(x) = 1/(2x - 3)

5.[10 pts] Use logarithmic differentiation to find the derivative.

y = x^{cos x}

6. [5 pts] If f(x) = x cos(x), find f ''(π/3).

7. [12 pts] Find the equation of the tangent line to f(x) = (2 + 3x)e^x at the point (0, 4)

8. [15 pts] Given a distance function for a particle s = 2t³ - 9t², t > 0.

a. Find the velocity and acceleration functions.

b. Graph the distance, velocity and acceleration functions for 0 < t < 6 on the same set of axes.

c. When does the particle come to a stop (i.e. v = 0)? When is the acceleration > 0 and when is the acceleration < 0?

9. [10 pts] Find the points on the circle x² + 9y² = 9 where the tangent line has slope 4.

10. [10 pts] Find the dy/dx of the following.

a. y = arcsin x

b. y = cot x
c.  y = x•cos x • e^x
d.  y = ln(x²e^x)

e.  y = cos(ln(2x))

Test 3

1. [9 pts] Two cars start moving from the same point. One travels north at 40 mi/h and the other travels east at 35 mi/h. At what rate is the distance btween the cars increasing 3 hours later?

2. [9 pts] Given f(x) = x³
a) For f(x) at a = 3, find L(x) = f(a) + f '(a)(x – a).

b) Use L(x) to approximate (3.01)³.

3. [10 pts] Find the critical numbers of f(x) = x ln(x).

4. [10 pts] Find the absolute maximum and the absolute minimum values of f(x) = x³ – x² + 5 on the interval [-1, 1].

5. [10 pts] Evaluate.

lim ( 3x²)/(sin x)

^x-->0 

6.[10 pts] Evaluate.

lim (sin x)^x

^x-->0

7. [5 pts] Given f(x) = 3x² - 2x – 1 on [-1/3, 1].

a) Is f(x) continuous on the interval?
b) Is f(x) differentiable on the interval?
c) Show that f(-1/3) = f(1).
d) Find the number c such that f '(c) = 0 on the interval.
e) What theorem did you use?

8. [12 pts] Given f(x) = x³ – 12x + 5.

a) Find the intervals where f(x) is increasing or decreasing.
b) Find the local maxima or local minima (if any)
d) Find the intervals where f(x) is concave up or concave down.
e) Find the inflection points (if any).

9. [10 pts] Find a positive number such that the difference of the number and the number squared is a maximum.

10. [15 pts] Given the function y = 2 – 15x – 9x² – x³.

A. Find the domain.
B. Find the x and y-int(s) (if any).
C. Find whether or not f(x) has even or odd symmetry.
D. Find the horizontal and vertical assymptotes (if any).
E. Find the intervals where f(x) is increasing or decreasing.
F. Find the points where f(x) has local maxima or local minima (if any).
G. Determine the concavity of f(x) and any inflection points.
H. Sketch the graph of f(x) using what you found in A-G.

Test 4

1. [10 pts] Find the most general antiderivative of the following:

a) f(x) = 3x – 4

b) f(x) = 4e^x + x

2. [9 pts] Find f, given f ''(x) = 2x + 5, f '(0) = - 4 and f(0) = 2

3. [10 pts] a) Sketch the graph of the function f(x) = 9 – x², -1 < x < 2.

b) Estimate the area under the graph of f(x) on [-1,2] using the LHS (left-hand sum) and

n = 3 rectangles.

4.  [10 pts] No calculations needed for this problem. Simply rewrite as indicated.

a) Rewrite the limit on [-1,5] as a definite integral.

              _n

lim          ∑ tan x_i ∆x =

^{n-->∞    i = 1}

b) Rewrite the integral as a limit of Riemann sums.

₂ ∫⁴ (x – 4e^x)dx =

                                                                        n

5. [10 pts] Use the equation _a ∫ ^_bf(x)dx = lim     ∑ f(x_i)∆x, where ∆x = (b – a)/n, to evaluate:

                                                             ^{n-->∞   i = 1}

₀ ∫⁴ (x – 4)dx =

6. [5 pts] a) Evaluate the definite integral.

₀ ∫⁴ (x – 4)dx =

b) Is your answer the same as #5? Why or why not?

7.[10 pts] Find the derivative of y.

a) y = ₄ ∫ ^x (x³ – 4x)dx

b) y = ₄ ∫^2x (x³ – 4x)dx

8. [10 pts] Evaluate the definite integral.

₀ ∫¹ (x – 4)(x + 2)dx 

9. [10 pts] Given y = x³.

a) Sketch the graph of y on [0, 2].
b) Shade the region to the right of the y-axis and to the left of y between y = 0 and y = 8
c) Find the area of the shaded region using a definite integral.

10. [16 pts] Evaluate the following indefinite integrals.

A. ∫ 3 sin x dx =
B. ∫ 1/x dx =
C. ∫ 5/x dx =
D. ∫ (x^1/2 - x^3/2) dx =