FINAL EXAM CALCULUS 1

This Exam Consists of 7 Sections:

C/ Graphs

D/ Max/Min

INSTRUCTIONS

This exam is 3 hours.
Student must stay for at least the first hour.
Student must answer all questions in the exam booklets provided.

.

.

Graphing Calculator not allowed. Show all your work.

1) Find the Limits

 a) b)  c) (9)

2) a) Discuss the discontinuities of b) Are any of the discontinuities removeable?
If so, what value for f(a) will remove the removeable discontinuity?

(8)

3) Given find values for c and d to make f(x) continuous over its entire domain.

(5)

4) Find the first derivative.Simplify only constants where possible.

a) y = log 7 (cos 5 3x) + ln (x 3 e 4 - x )
b) y = -5 arcsin (x 4 + 3x) - 3 arcsec (2 - x 2 ) + (7) 4x + 3 - 19
c) (10)

5) Differentiate implicitly: x sin y = y cos x

(3)

6) a) Use logarithmic differentiation to find y ' if .

b) Write the equation of the tangent to this curve at the point where x = -2

(4)

.

7) The position function s(t) of point P on a coordinate line is: , t in seconds

a) Find v(t) the velocity function.

b) At what value(s) of t is the point

 i) moving right? ii) moving left? iii) not moving?

c) Find a(t) the acceleration function.

d) At what value(s) of t is the point

 i) accelerating? ii) decelerating? iii) moving at a constant speed?

.

(8)

8) Find the oblique and vertical asymptotes for (4)

9) Apply the 1st and 2nd derivative tests to find the critical and inflection points,
increasing, decreasing, concave up and concave down intervals, for , and sketch the graph.

(15)

10) Find the dimensions of the right circular cylinder of maximum volume that can be
inscribed in a cone with height 15 cm and base radius 7 cm. (make a diagram!)

(5)

11) Use differentials to approximate the maximum error in the
calculated
volume of a sphere if the radius measures 12 inches
with an error in measurement of
! 0.06 inches.

(4)

12) A point P(x, y) moves along the graph of y 2 = x 2 - 9
such that dx/dt = 1/x units/sec. Find dy/dt at the point (5, 4).

(4)

.

13) A civil engineer is designing a "sound-blocking wall" for a
housing development situated near a highway curve.
The curve follows the rule The diagram is a scaled down version of the situation.
Find the coordinates of T, the point to build the wall for maximum
sound-blocking effect. The wall must be parallel to the line PQ. (5)

14) Find the most general antiderivative of

 a) b) c) (hint: divide)

(6)

15) Solve these differential equations subject to the given conditions:

 a) f '"(x) = 6x, f "(0) = 2, f '(0) = -1, f(0) = 4 b) f "(x) = x + cos x, f '(0) = 2, f(0) = 1

(6)

16) A point moves on a coordinate line with a(t) = 2 - 6t.
If the initial conditions are v(0) = -5, and s(0) = 4, find s(t).
{reminder: if s(t) is the position function, v(t) = s '(t) and a(t) = s "(t).}

(4)

Total 100

.

.

.