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) = kx2+2 If x > -1

(7)Consider the formula s(t) = t2 + 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 second-order derivative for . Do not simplify.

[3] (10) Find all x-values 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 cm3. 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 - x2 and the x-axis.