﻿ Slope Tangent Worksheet | Problems & Solutions
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# Slope Tangent Worksheet

Slope Tangent Worksheet
• Page 1
1.
If $f$($x$) = 5$x$2 + 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 $f$($x$) = 28$x$3, find $f$ ′($x$).
 a. 3$x$2 b. 84$x$2 c. 28$x$2 d. 31$x$2

#### 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 $f$($x$) = $\frac{35}{{x}^{2}}$.
 a. - $\frac{35}{{x}^{3}}$ b. - $\frac{70}{{x}^{3}}$ c. - $\frac{70}{{x}^{2}}$ d. - $\frac{35}{{x}^{2}}$

#### 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(4$x$) from the following.
 a. $\frac{1}{x}$ b. c. -

#### 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 $f$ ($x$) = $x$2 + 2$x$ + 5?
 a. 2$x$2 - 10 b. 2$x$2 + 10 c. 2$x$ + 2 d. 2$x$ - 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 $f$ ($x$) = $x$3 + $x$2.
 a. 3$x$2 - 2$x$ b. 3$x$2 + 10 c. 2$x$2 + $x$ d. 3$x$2 + 2$x$

#### 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 $f$ ($x$) = 5$x$ - 8$x$2 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 $f$ ($x$) = 17$x$ - $x$2 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 $f$($x$) = $x$2 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 $f$($x$) = | 5$x$ + 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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