Name___________________________
AP Calculus AB
Graded Assignment #1
All work should be completed neatly on a separate sheet of paper. Final numeric or derivative answer on answer sheet.
1. Use the limit definition of the derivative to find f '(x) if f(x) =
2. Explain the meaning of the expression limℎ→0
3. y = 5 x 2 +
2
, Find y’(-1)
x
4. Find lim
y 2 − sin(2 y )
tan(5 y )
y →0
𝜋
6
sin� +ℎ�−
ℎ
1
2
3
.
x2
. What is the exact value of this expression?
sin 4 x
x →0 sin 6 x
5. lim
6.
Given: P(x) = (4x + 6)(f(x) + 6x) , f(3) = − 3 and f '(3) = 10.
7.
Find the equation of the tangent line to the curve f(x) = 2 x 3 + 5 x − x + 8 at x = 1.
1
Let f (t ) = for t > 0 . For what value of t( in terms of a and b) is f ' (t ) equal to the average rate of
t
change of f(t) on the closed interval [a, b]?
8.
Find: P '(3).
9.
108 x 5 + 106 x 4 + 10 4 x 2
.
Find lim 9 4
x→∞ 10 x + 10 7 x 5 + 105 x 3
10.
 x 2 + bx
,x ≤ 5

If f is a continuous function defined by f ( x) = 
, then find b.
π
5 sin( x) , x > 5
2

11.
Find lim
12.
Find lim
13.
If f ( x) = e3 ln( x ) , then find f ' ( x) . Answer must be completely simplified.
9 − 6x + x2
.
x→3 (6 − 2 x )
x→b
b−x
.
x− b
2
14. If f(x) = (3x – 2)3 what is the value of the (f-1 ) ‘ (x) at x = 8
15. Textbook pg 225 #30
16. Textbook pg 225 # 41
17-25 Find derivative
17.
19. f ( x) =
21.
18. f ( x) =
f ( x) = e 2 x + 1
( x + 1) 6 (3 x 2 − 4) 2 / 3
x −1
xy + y 2 − x 3 =
5
x −1
20. f ( x) = [sin(2 x)]cos x
4x
+ 16; Find dy/dx at (1,4)
y
22. Find dy / dx if y2 +3xy = ln(xy) + 5cos-1 (2x)
23. y = sin-1(ln(x2/3))
24. F(x) = 3cos(5x)
25. V(t) = (log3((4t+1)8))5
26. Y = ln[csc(3x)] ecsc(3x)
27. Let f be the function given by f ( x) = x 2 ln( x) . For what value of x = c is the slope of the line tangent to the
graph of f(x) at x=c equal to 2?
28. Find the derivative of the inverse of f(x) =x3 – x – 2 at x = 1
29. At what point(s)(x,y) does the curve y 2 − x 3 − 15 x 2 = 0 have horizontal tangent lines?
30. At what point(s) (x, y) on the curve x 3 − y 2 + x 2 = y is the tangent line vertical?
31 - 36. Utilize the chart below
X
-2
-1
1
2
f(x)
16
2
1
-1
g(x)
2
-1
5
3
f'(x)
8
-2
6
3
Find the derivative of the functions below at the indicated x value
𝑔(𝑥)
31. f(x)g(x) at x = -2
32.
33. �𝑓(𝑥) at x = -2
34. (2x+5)(g(x))+ (f(x) + g(x))4 at x = 2
35. f-1(x) at x = 2
𝑓(𝑋)
at x = -1
36. g-1(x) at x = 3
g'(x)
-1
3
2
8