Calculus 1 Winter 1997
[6] (1) Refer to the sketch below to answer the following questions.
If a limit does not exist, state in which way (+ infinity,  infinity,
or "does not exist").
(a) ____ (b) ____
(c) ____ (d) ____
(e) ____ (f) ____
(g) ____ (h) ____
(i) ____ (j) ____
(k) ____
(l) _____ is one value of x for which f is continuous and at the same time f is not differentiable.
[10] (2) Use algebraic techniques to calculate the following limits. If an answer is undefined, assign the symbol + or  if possible.
b)
[3]
(3) (a) Compute the table below for
x 
0.1 
0.01 
0.001 
0.001 
0.01 
0.1 
f(x) 
(b) Use your results to give = _________.
[4] (4) Given:
Use the definition of continuity at a point to:
(a) determine whether or not f is continuous at x = 1.
(b) determine whether or not f is continuous at x = 1.
[3] (5) Given:
(a) Find the equations for all vertical asymptotes for g(x).
(b) Find any other discontinuities for g(x).
[2] (6) Find a value for k which will make f continuous at x = 1.
f(x) = 3  2x If x less than or equal 1
f(x) = kx^{2}+2 If x > 1
(7)Consider the formula s(t) = t^{2} + t in which t is measured in seconds and s(t) is measured in centimetres.
[2] (a) Determine the average velocity from t = 3 to t = 4.
[3] (b) Use a limit definition to find the derivative s'(t).
[2] (c) What is the instantaneous velocity at t = 3?
(8) Give the derivatives of the following functions. Do not simplify your answers.
[3] (a)
[3] (b)
[3] (c) y = sin(2x) sec(3x)
[3] (d)
[3] (e)
[4] (f) (Use logarithmic differentiation)
[4] (9) Find the secondorder derivative for . Do not simplify.
[3] (10) Find all xvalues at which the graph of f has a horizontal tangent line.
(11) Given
[3] (a) Find:
[2] (b) Find an equation of the tangent line at (1, 1).
[3] (12) Find the maximum and minimum values of f on the given closed interval and state where these values occur. Justify your answer.
[5] (13) A box with a square base and open top must have a volume of 32 000 cm^{3}. Find the dimensions of the box that minimize the amount of material used.
[5] (14) Sketch the graph of a function f having the following characteristics.
for and for
for and for
for
for
[5] (15) By referring to the graph, complete the chart below by writing in each blank space one of the following symbols: + (positive),  (negative), 0 (zero), or (does not exist).
x < 3 
x = 3 
3 < x < 0 
x = 0 
0 < x < 4 

f(x) 

f'(x) 

f"(x) 
(16) Evaluate the following integrals.
[2] (a)
[2] (b)
[2] (c)
[3] (d)
[3] (e) Show all work
[4] (17) Find the area of the region bounded by the graph of y = 4  x^{2} and the xaxis.