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*.

5)

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 + x^{2})(2x - 5) |
c) y = -6x^{ }^{½} (x^{3} + 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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