Name______________________________________
Calculus
Period_____
Review 2.4, 3.1-3.5
2
1. The curve y  ax  bx  c passes through (1,10) and is tangent to y  4 x  3 at the y-intercept. Find a, b,
and c.
2. Suppose that y  5 x  2 is the equation of the tangent line to the graph of y  f  x  at x = 3. What is f  3 ?
What is f   3 ?
Suppose f  2  1, f   2  4, g  2  5, and g  2   3. Find the derivative at 2 of each of the following
functions.
f ( x)
3. p( x)  2 f ( x)  g ( x)
4. r ( x) 
g ( x)
5. q( x)  f ( x)  g ( x)
6. t ( x) 
f ( x)
f ( x)  g ( x)
7. If velocity is negative and acceleration is positive, then speed is ______________________.
8. If velocity is positive and speed is decreasing, then acceleration is ______________________.
9. If velocity is positive and decreasing, then speed is ________________________.
10. If speed is increasing and acceleration is negative, then velocity is _______________.
11. If velocity is negative and increasing, then speed is ____________________.
12. If the particle is moving to the left and speed is decreasing, then acceleration is ____________________.
13. A particle moves along the x-axis so that the position at any time t  0 is given by x  t   t 3  t 2  t  3. For
what values of t, 0  t  3 is the particle’s instantaneous velocity the same as its average velocity on the closed
interval [0,3]?
14. A particle moves along the x-axis so that its position in feet at any time t  0 is given by x  t   t 4  2t 2  4 .
a) Find an expression for the velocity of the particle at any time t  0 .
b) Find the average velocity of the particle for the first two seconds.
c) Find the instantaneous velocity of the particle at t = 2 seconds.
d) Find the values of t for which the particle is at rest.
e) Find the position of the particle when it is at rest.
f) Find the displacement of the particle from t = 0 to t = 3 seconds.
g) Find the total distance traveled by the particle from t = 0 to t = 3 seconds.
Differentiate.
15. y  4 x3  7 x 2  22
16. y  (5 x  3) sec( x)
17. f ( x)  4 x  3 x5
18. y  cos x sin x
19. y  ( x  5)(2 x  1)(2 x  1)
20. y 
21. y  5 csc( x)
22. y  x  tan( x)
23. y 
cot x
3 x  sec x
2x  5
x2  1
24. Show that the graphs of y = tanx and y = cotx have no horizontal tangents.
25. The line that is normal to the cure y = x2 at the point (1,1) intersects the curve at what other point?