Cal I Xmas Review

Review of Topics Covered in a 2-Semester Differential Calculus Course.

Show all your work.

1) f(x) = x 3 - x 2 - 4x + 4

a) Find the zeros of f(x).

b) Find the equation of the tangent to f(x) at P(0, 4).

c) Find the point(s) on f(x) where the tangent is parallel to the line x - y + 7 = 0.

d) Use the first derivative to find all critical points and to list the increasing
and decreasing intervals of f(x). Label the critical points maximums or minimums.

e) Find the absolute and local extrema for f(x) on the interval [0, 3]

2) Sketch the graph and discuss the properties of g(x) = 2x 4 - 8x 2.

3) We're making a cardboard box with an open top, a square base,
with volume = 4 dm 3. What dimensions will use a minimum of cardboard?

4) y = x 2 + 2x .

a) Using the limit definition of the derivative, find the slope of the tangent to y at any point.

b) Using your answer from (a), find the slope of the tangent to y at (-3, 3).

c) Using the rules of derivatives, find the equation of the tangent to y at the point where x = 4.


a) What are the domain and range of y?

b) Find dy/dx using first principles.

c) Write the equation of the tangent to y at P(7, ½ ).

6) Use the power, product and quotient rules to differentiate:

a) b) y = (6 - 3x + x2)(2x - 5) c) y = -6x ½ (x3 + 7x - 1)
d) g(x) = (7x 5/3 + 9)(2 - 6x 7) e) f)

7) The position of a particle that is moving linearly at time t is:

s(t) = t 3 - 6t 2 + 9t

where s(t) is in meters and t is in seconds.

a) Find the velocity at time t.

b) What is the velocity after 2 seconds? after 4 seconds?

c) When is the particle at rest?

d) When is the particle moving in a positive direction?

e) Draw a diagram to show the motion of the particle.

f) What is the total distance traveled during the first 5 seconds?


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