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Slope Tangent Worksheet

Slope Tangent Worksheet
  • Page 1
 1.  
If f(x) = 5x2 + 8, then find f′(2).
a.
5
b.
15
c.
25
d.
20


Solution:

f ′(2) = limh0f(2 + h) - f(2)h

= limh05(2 + h)2 + 8 - [5(22) + 8]h

= limh05(4 + 4h +h2) + 8 - 20 - 8h

= limh0 5h2 + 20hh

= limh0 (5h + 20)
[hh = 1, since h ≠ 0.]

= 20


Correct answer : (4)
 2.  
If f(x) = 28x3, find f ′(x).
a.
3x2
b.
84x2
c.
28x2
d.
31x2


Solution:

f ′(x) = limh0 f(x + h) - f(x)h

= limh028(x + h)3 - 28x3h

= 28limh0 (x + h - x)((x + h)2 + x(x + h) +x2)h
[Factor the numerator.]

= 28limh0 ((x+ h)2 + x(x + h) + x2)
[hh = 1, since h ≠ 0.]

= 28(3x2) = 84x2


Correct answer : (2)
 3.  
Find f ′(x), if f(x) = 35x2.
a.
- 35x3
b.
- 70x3
c.
- 70x2
d.
- 35x2


Solution:

f ′(x) = limh0 f(x + h) - f(x)h
[Definition.]

= limh0 35(x + h)2-35x2h

= limh0 35h [1(x + h)²-1x²]

= limh0 35h [x²-(x + h)²x²(x + h)²]

= limh0 35h [- 2hx - h²x²(x + h)²]

= limh0 - 35(2x + h)x²(x + h)²
[hh = 1, since h ≠ 0.]

= - 70x³


Correct answer : (2)
 4.  
Choose f ′(x), for f(x) = 46ln(4x) from the following.
a.
1x
b.
46 x
c.
- 46 x


Solution:

f ′(x) = limh0 f(x + h) Ã¢â‚¬â€œ f(x)h
[Definition.]

= limh0 46ln4(x + h) - 46ln(4x)h

= 46limh0 ln (4(x + h)4x)1h

= 46limh0 ln (1 + hx)1h

= 46limh0 ln [(1 + hx)xh]1x

= 46ln [limh0(1 + hx)xh]1x

= 46(ln e1x)
[limh0 (1 + x)1x = e.]

= 46x ln e

= 46x
[ln e = 1.]


Correct answer : (2)
 5.  
Which of the following is the derivative of f (x) = x2 + 2x + 5?
a.
2x2 - 10
b.
2x2 + 10
c.
2x + 2
d.
2x - 2


Solution:

f ′(x) = limh0 f(x + h) Ã¢â‚¬â€œ f(x)h
[Definition.]

= limh0 (x + h)2 + 2(x + h) + 5 - (x2 + 2x + 5)h

= limh0 x2+2hx+h2 + 2x + 2h + 5-x2 - 2x - 5h

= limh0 2hx + 2h+h2h

= limh0 (2x + 2 + h)
[hh = 1, since h ≠ 0.]

f ′(x) = 2x + 2


Correct answer : (3)
 6.  
Find the derivative of f (x) = x3 + x2.
a.
3x2 - 2x
b.
3x2 + 10
c.
2x2 + x
d.
3x2 + 2x


Solution:

f ′(x) = dydx = limh0 f(x + h) Ã¢â‚¬â€œ f(x)h
[Definition.]

= limh0 (x + h)3+(x + h)2-x3-x2h

= limh0 (x + h)3-x3+(x + h)2-x2h

= limh0 (x + h - x)[(x + h)2+x(x + h)+(x)2] + (x + h - x)(x + h + x)h

= limh0 h[(x + h)2+x(x + h)+x2+2x + h]h

= limh0 [(x + h)2 + x(x + h) + x2 +2x + h]
[hh = 1, since h ≠ 0.]

= 3x2 + 2x

f ′(x) = 3x2 + 2x


Correct answer : (4)
 7.  
Find the derivative of f (x) = 5x - 8x2 at the point x = 3.
a.
- 48
b.
- 53
c.
- 8
d.
- 43


Solution:

f ′(x) = limh0 f(3 + h) Ã¢â‚¬â€œ f(3)h
[Use the Definition.]

= limh0 5(3 + h) - 8(3 + h)2 - [5(3) - 8(3)2]h

= limh0 15 + 5h - 8(9+h2+6h) - 15 + 72h

= limh0 - 8h2 - 43hh

= limh0 (- 8h - 43)
[hh = 1, since h ≠ 0.]

= - 43


Correct answer : (4)
 8.  
Find the slope of the tangent to the curve f (x) = 17x - x2 at x = 4.
a.
- 17
b.
17
c.
- 9
d.
9


Solution:

Given, f (x) = 17x - x2.

= limh0 (17(x+h)-(x+h)2)-(17x-x2)h
[f′ (x) = limh0 f(x+h)-f(x)h.]

= limh0 17h-h2-2xhh = limh0 17 - h - 2x

= 17 - 2x

At x = 4, the slope of the tangent to the curve = limx4 f′ (x) = limx4 17 - 2x

= 17 - 2(4) = 9


Correct answer : (4)
 9.  
Find the slope of the graph of the function f(x) = x2 at x = - 9.
a.
9
b.
- 18
c.
- 9


Solution:

At x = - 9, the slope of the tangent to the curve = limx- 9 x2-(- 9)2x + 9
[Use the definition.]

= limx- 9 (x - 9)(x + 9)x + 9

= limx- 9 (x - 9)

= - 18


Correct answer : (2)
 10.  
Estimate the slope of the tangent line to the curve f(x) = | 5x + 44| at x = - 3.
a.
5
b.
- 5
c.
- 3
d.
1


Solution:

At x = - 3, the slope of the tangent line to the curve = limx- 3 - (5x + 44) -  ( | 5(- 3) + 44| )x + 3
[At x = - 3, (5x + 44) is negative.]

= limx- 3 - 5(x + 3)(x + 3) = - 5


Correct answer : (2)

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