Derivative with the Tangent Line Problem Worksheet

Derivative with the Tangent Line Problem Worksheet
  • Page 1
 1.  
Find the slope of the curve y = x5x+8 at the point x = 2.
a.
8 9
b.
4 9
c.
1 5
d.
7 81
e.
2 81


Solution:

y = x5x+8
[Equation of the curve.]

dydx = ddx (x5x+8)
[Find dydx.]

= (5x+8)(1)-x(5)(5x+8)2
[Use Quotient Rule.]

= 8(5x+8)2
[Simplify.]

(dydx)x=2 = 8(5(2)+8)2
[Find the slope of the curve at x = 2]

= 2 / 81
[Simplify.]

So, the slope of the curve at x = 2 is 2 / 81.


Correct answer : (5)
 2.  
Find the slope of the curve y = 6x2 + 7x at x = 3.
a.
36
b.
25
c.
43
d.
387
e.
75


Solution:

y = 6x2 + 7x
[Equation of the curve.]

dydx = ddx (6x2 + 7x)
[Find dydx.]

dydx = 12x + 7
[Use Sum Rule.]

(dydx)x = 3 = 12(3) + 7 = 43
[Find the slope of the curve at x = 3]

So, the slope of the curve at x = 3 is 43.


Correct answer : (3)
 3.  
Find the slope of the curve y = 6x3 at x = 2a.
a.
48a4
b.
24a2
c.
96a4
d.
3a2
e.
72a2


Solution:

y = 6x3
[Equation of the curve.]

dydx = ddx (6x3)
[Find dydx.]

dydx = 18x2

(dydx)x = 2a = 72a2
[Find the slope of the curve at x = 2a]

So, the slope of the curve at x = 2a is 72a2.


Correct answer : (5)
 4.  
Find the slope of the curve y = 6x+33x2+6x+4 at x = 1.
a.
42 169
b.
9 13
c.
- 30 169
d.
2 3
e.
- 132 169


Solution:

y = 6x+33x2+6x+4
[Equation of the curve.]

dydx = ddx (6x+33x2+6x+4)
[Find dydx.]

dydx = (3x2+6x+4)(6)-(6x+3)(6x+6)(3x2+6x+4)2
[Use Quotient Rule.]

= - 18x2 - 18x+6(3x2+6x+4)2
[Simplify the numerator.]

(dydx)x = 1 = -18-18+6169
[Find the slope of the curve at x = 1]

= - 30 / 169
[Simplify.]

So, the slope of the curve at x = 1 is - 30 / 169.


Correct answer : (3)
 5.  
Find the slope of the curve y = 98x at x = 4.
a.
18
b.
9 32
c.
- 9 128
d.
72


Solution:

y = 98x
[Equation of the curve.]

dydx = ddx (98x)
[Find dydx.]

dydx = 8x(0) - 9(8)64x2
[Use Quotient Rule.]

= - 7264x2
[Simplify the numerator.]

(dydx)x = 4 = - 7264(4)2 = - 9 / 128
[Find the slope of the curve at x = 4]

So, the slope of the curve at x = 4 is - 9 / 128.


Correct answer : (3)
 6.  
Find the slope of the curve y = 7x2 at x = 4.
a.
49 32
b.
7 32
c.
116
d.
- 7 4
e.
- 7 32


Solution:

y = 7x2
[Equation of the curve.]

dydx = ddx (7x-2)
[Find dydx.]

dydx = - 14x- 3 = - 14x3
[Use Power Rule.]

(dydx)x = 4 = - 14(4)3 = - 7 / 32
[Find the slope of the curve at x = 4]

So, the slope of the curve at x = 4 is - 7 / 32.


Correct answer : (5)
 7.  
Find the slope of the curve y = e2x + 5 at x = 3.
a.
e32
b.
2e11
c.
11e10
d.
e11
e.
e10


Solution:

y = e2x + 5
[Equation of the curve.]

dydx = ddx(e2x + 5)
[Find dydx.]

dydx = 2e2x + 5
[Use Chain Rule.]

(dydx)x = 3 = 2e11
[Find the slope of the curve at x = 3]

So, the slope of the curve at x = 3 is 2e11.


Correct answer : (2)
 8.  
Find the slope of the curve y = 4x+56x+7 at x = 2.
a.
13 19
b.
2 3
c.
- 154 19
d.
- 2 361
e.
154 361


Solution:

y = 4x+56x+7
[Equation of the curve.]

dydx = ddx (4x+56x+7)
[Find dydx.]

dydx = (6x+7)(4)-(4x+5)(6)(6x+7)2
[Use Quotient Rule.]

= 24x+28-24x-30(6x+7)2
[Expand the numerator.]

= - 2(6x+7)2
[Simplify.]

(dydx)x = 2 = - 2(6(2)+7)2 = - 2361
[Find the slope of the curve at x = 2]

So, the slope of the curve at x = 2 is - 2 / 361.


Correct answer : (4)
 9.  
Find the slope of the curve y = x + 25 at x = 3.
a.
1
b.
18 5
c.
15
d.
13
e.
5


Solution:

y = x+25
[Equation of the curve.]

dydx = 1 / 5ddx(x+2)
[Find dydx.]

= 15(1 + 0) = 15
[Use Sum Rule.]

(dydx)x = 3 = 15
[Find the slope of the curve at x = 3.]

So, the slope of the curve at x = 3 is 1 / 5.


Correct answer : (3)
 10.  
Find the slope of the curve y = ln(3x + 11) at x = 3.
a.
ln (20)
b.
1 20
c.
ln (38)
d.
3 20
e.
27 20


Solution:

y = ln(3x + 11)
[Equation of the curve.]

dydx = ddx (ln(3x + 11))
[Find dydx.]

dydx = 33x+11
[Use Chain Rule.]

(dydx)x = 3 = 33(3)+11 = 3 / 20
[Find the slope of the curve at x = 3]

So, the slope of the curve at x = 3 is 3 / 20.


Correct answer : (4)

*AP and SAT are registered trademarks of the College Board.