Deck 1: The Derivative

Question 1

Find the equation of the following line: Slope is 3; y-intercept is 5.
Enter your answer in slope-intercept form.

Answer 1

y = 3x + 5

Question 2

Find the equation of the following line: Perpendicular to 5x = 7 - 8y; y-intercept is -2.
Enter your answer in slope-intercept form with any fractions in the form in lowest terms.

Answer 2

y = x - 2

Question 3

Find the slope of the line with the equation y = 4 - 5x.
Enter just an integer.

Answer 3

-5

Question 4

Find the slope of the line with equation 3y + 2 = 5x - 2y?

A) - $\frac { 1 } { 2 }$
B) $\frac { 5 } { 3 }$
C) 5
D) 1

Question 5

Find the equation of the following line: and on the line (in the xy -plane).
Enter your equation in the simplest possible form.

Question 6

Find the equation of the following line: and (2, 0) on line.
Enter your answer in point-slope form using .

Question 7

Find the equation of the following line: Parallel to y = - $\frac { 1 } { 4 }$ x + 8; (4, 4) on line.

A) y = - $\frac { 1 } { 4 }$ x + 5
B) y = - 4x - 20
C) y = $\frac { 1 } { 4 }$ x - 5
D) y = - $\frac { 1 } { 4 }$ x - 5

Question 8

Find the slope of the line with the equation 2x + 4y = 5.
Enter just a fraction of form in lowest terms.

Question 9

Find the slope of the line with equation 2x + 4 = 2(2y + 3).
Enter just a fraction of form in lowest terms.

Question 10

Find the equation of the following line: Slope is - ; on line.
Enter your answer in point-slope form.

Question 11

Which of the following is/are true of the lines 2x - 5y = -15 and 5x + 2y = 6? (I) They are parallel.
(II) They are perpendicular.
(III) They cross the x-axis at the same point.
(IV) They cross the y-axis at the same point.

A) I
B) II and III
C) II
D) II and IV
E) none of these

Question 12

Find the equation of the following line: (2, 3) and (4, 6) on line.
Enter your answer in slope-intercept form.
Include both the slope and the intercept in your equation.

Question 13

Find the equation of the following line: Perpendicular to - 8x + y - 3 = 0; (4, 3) on line.

A) y = - $\frac { 1 } { 8 }$ x - $\frac { 7 } { 2 }$
B) y = - 8x - 28
C) y = - $\frac { 1 } { 8 }$ x + $\frac { 7 } { 2 }$
D) y = $\frac { 1 } { 8 }$ x - $\frac { 7 } { 2 }$

Question 14

Find the equation of the following line: Perpendicular to y = $\frac { 1 } { 5 }$ x + 8; (2, 3) on line.

A) y = - 5x + 13
B) y = 5x - 13
C) y = - $\frac { 1 } { 5 }$ x - $\frac { 13 } { 5 }$
D) y = - 5x - 13

Question 15

Consider 2 lines each having slope $\frac { 4 } { 3 }$ , one passing through the point (-1, 2) and the other through (3, 5). Which of the following is/are true?
(I) They are parallel.
(II) They are the same line.
(III) They cross the x-axis at the same point.
(IV) They intersect at the point (0, 3).

A) I
B) IV
C) II
D) III
E) none of these

Question 16

Find the equation of the following line: Perpendicular to 3y - x = 1; y-intercept is .
Enter your answer in slope-intercept form.

Question 17

Find the equation of the following line: Parallel to y = - 9x + 4; (2, - 2) on line.

A) y = 9x - 16
B) y = - 9x - 16
C) y = - 9x + 16
D) $y = - \frac { 1 } { 9 } x - \frac { 16 } { 9 }$

Question 18

Each of the lines (I), (II), (III), and (IV) in the figure below is the graph of one of the equations (i), (ii), (iii), and (iv). Match each equation with its graph. (a) x + y = 3 (b) x - y = 3 (c) x + y = -3 (d) x - y = -3

A)
(a) IV
(b) III
(c) II
(d) I
B)
(a) II
(b) III
(c) I
(d) IV
C)
(a) I
(b) II
(c) III
(d) IV
D)
(a) III
(b) II
(c) IV
(d) I

Question 19

Find the equation of the following line: Parallel to $6 x + y - 3 = 0$ ; (4, 3) on line.

A) y = - $\frac { 1 } { 6 }$ x - $\frac { 9 } { 2 }$
B) y = 6x - 27
C) y = - 6x - 27
D) y = - 6x + 27

Question 20

-The average value of a certain type of automobile was $15,780 in 1994 and depreciated to $5400 in 1999. Let y be the average value of the automobile in the year x, where x = 0 represents 1994. Write a linear equation in slope-intercept form that models the value of the automobile, y, x years after 1994.

A) y = - $\frac { 1 } { 2076 }$ x - 5400
B) y = -2076x + 5400
C) y = -2076x - 4980
D) y = -2076x + 15,780
Referring to the graph above , assign one of the following descriptors to the point: large positive slope, small positive slope, zero slope, small negative slope, large negative slope.

Question 21

Find all points on the graph of y = where the curve has slope 12. The slope of the tangent line to the graph is 3 .
Enter your answer exactly in the form: (a, b), (c, d) where a > c.

Question 22

If (x,y) is a point on the parabola y = 3 $x ^ { 2 }$ , then the tangent line to y = 3 $x ^ { 2 }$ passing through (x,y) has slope 6x. Find the equation of the line tangent to y = 3 $x ^ { 2 }$ through the point (-2, 12).

A) y = 6x + 12
B) y = 12x + 6
C) y = -12x - 12
D) y = -2x + 12

Question 23

Which of the following lines is/are parallel to the tangent line of the graph of $y = - x ^ { 3 }$ at the point where $x = 1$ and the slope of the tangent line is $- 3 x ^ { 2 }$ .
(I) y = -3x + 1
(II) y = 3x + 1
(III) y = -3x - 4
(IV) y = 3x + 1

A) (II) and (III)
B) (I) and (III)
C) (I) and (II)
D) (III) and (IV)
E) none of these

Question 24

S

A) large positive slope
B) large negative slope
C) small negative slope
D) zero slope

Question 25

Find the derivative of f(x) = 7x - 19 at x = 2.

A) 9
B) 14
C) 34
D) 7

Question 26

Find the derivative of f(x) = .
Enter your answer exactly in the form: a

Question 27

Find the derivative of f(x) = .
Enter your answer exactly in the form: where a, b are integers. Do not use parentheses.

Question 28

Find the derivative of f(x) = 4 .
Enter your answer in the form: , where a, b are either fractions in lowest terms or integers. No parentheses.

Question 29

Find the derivative of f(x) = 2 $\sqrt { x }$ at x = 4.

A) - $\frac { 1 } { 2 }$
B) $\frac { 1 } { 2 }$
C) -1
D) 2

Question 30

Find the slope of the tangent line to the graph of y = at the point (0.9, 0.81) and write the equation of the tangent line.
Enter your answer exactly as just the equation in standard point-slope form.

Question 31

P

A) large positive slope
B) small negative slope
C) small positive slope
D) large negative slope

Question 32

R

A) small positive slope
B) small negative slope
C) zero slope
D) large negative slope
Solve the problem.

Question 33

Find the slope of the curve y = $x ^ { 6 }$ at x = -3.

A) slope = -4374
B) slope = -1458
C) slope = 1458
D) slope = 4374

Question 34

Find the derivative of f(x) = .
Enter your answer exactly in the form: a . Do not use parentheses.
No parentheses.

Question 35

Find the derivative of $f ( x ) = \frac { 1 } { \sqrt [ 3 ] { x } }$

A) - $\frac { 1 } { 3 x ^ { 4 / 3 } }$
B) $\frac { 2 } { 3 }$ $x ^ { - 4 / 3 }$
C) $\frac { 5 } { 3 x ^ { 2 } }$
D) none of these

Question 36

Consider the curve f(x) in the accompanying sketch. Find the slope of the tangent line at the point where x = 2.
Enter your answer as just a fraction or an integer.

Question 37

Find the derivative of f(x) = $x ^ { 5 / 2 }$ .

A) 5 $x ^ { 3 / 2 }$
B) $\frac { 5 } { 2 }$ $x ^ { - 1 / 2 }$
C) $\frac { 5 } { 2 }$ $x ^ { 3 / 2 }$
D) $\frac { 2 } { 5 }$ $x ^ { - 1 / 2 }$

Question 38

Q

A) large negative slope
B) small negative slope
C) zero slope
D) small positive slope

Question 39

Find the slope of the tangent line to the graph of y = at the point and write the equation of the tangent line.
Enter your answer exactly as just the equation in standard point-slope form. Use all fractions of form in lowest terms.

Question 40

Find all points at which the tangent line to the graph of y = is parallel to the line .
Enter your answer in the form: (a, b), (c, d)where a > c

Question 41

If g(t) = - + 2t - 3, find and simplify.
Enter your answer exactly as a polynomial in a in standard form.

Question 42

Calculate the following limit(s) if they exist.
[ + 1) ]
Enter your answer as an integer, or " does not exist"

Question 43

Calculate the following limit(s) if they exist.
( - 2x + 5).
Enter just an integer or "does not exist".

Question 44

Which of the following is the equation of the line tangent to the curve $y = \sqrt { x }$ at $x = 9$ ?
(I) y = -x - $\frac { 1 } { 6 }$
(II) y = $\frac { 1 } { 2 }$ x + 3
(III) y = $\frac { 1 } { 6 }$ x + $\frac { 3 } { 2 }$
(IV) y = $\frac { 1 } { 2 }$ $x ^ { - 1 / 2 }$ - $\frac { 1 } { 6 }$

A) IV
B) III
C) I
D) II
E) none of these

Question 45

If f(x) = 2 $x ^ { 2 }$ + 5x, find $\frac { f ( x + h ) - f ( x ) } { h }$ and simplify.

A) 4x + 5h + 4 $\mathrm { h } ^ { 2 }$
B) 4x + 5xh + $\mathrm { h } ^ { 2 }$ + 5h
C) 4x + 5 + 4 $\mathrm { h } ^ { 2 }$ + 5h
D) 4x + 5 + 4h
E) none of these

Question 46

If f(x) = -2 $x ^ { 2 }$ - 3x + 2, find $\frac { f ( x + h ) - f ( x ) } { h }$ where h ? 0.

A) $\frac { - 2 x ^ { 2 } - 2 h ^ { 2 } - 3 h } { h }$
B) $\frac { - 2 h ^ { 2 } - 3 h + 2 } { h }$
C) -4x - 2h - 3
D) -4x - 1
E) none of these

Question 47

If g(x) = 2 - x + 3, compute g(1) and g'(1).
Enter your answer exactly in the form: g(1) = a, g'(1) = b
where a,b are integers

Question 48

Determine: if y = 4 - 6x.
Enter your answer as just an integer.

Question 49

Calculate the following limit(s) if they exist.
(2x + 2 Enter your answer as an integer, or "does not exist".

Question 50

If f(t) = 5t - 4, find where h ≠ 0 and simplify.
Enter your answer as just an integer.

Question 51

Calculate the following limit(s) if they exist.
4 Enter your answer as an integer.

Question 52

Find the derivative of f(x) = at x = 32.
Enter just a reduced fraction or an integer.

Question 53

Calculate the following limit(s) if they exist.
Enter your answer as a fraction, integer, or "does not exist".

Question 54

What is the equation of the tangent line to the graph of y = at the point where ?
Enter your answer in standard point-slope form.

Question 55

Calculate the following limit(s) if they exist.
Enter your answer as just a fraction or an integer, or the words " does not exist"

Question 56

Calculate the following limit(s) if they exist.
Enter your answer as just a fraction in lowest terms or an integer, or the words "does not exist".

Question 57

Calculate the following limit(s) if they exist.
Enter your answer as just an integer or a fraction in lowest terms, or the words "does not exist".

Question 58

Determine: if y = .
Enter your answer as just an integer or a reduced fraction.

Question 59

If f(x)= - $x ^ { 2 }$ + 2x - 1, find $\frac { f ( x + h ) - f ( x ) } { h }$ , h ? 0.

A) -2x + 2
B) -2xh - $x ^ { 2 }$ + 2h
C) $\frac { \mathrm { h } ( - \mathrm { h } + 2 ) } { \mathrm { h } }$
D) $\frac { h ( - 2 x - h ) } { h }$
E) -2x - h + 2

Question 60

Find the derivative of f(x) = .
Enter your answer in the form: a , where a, b are either fractions of the form or integers. No parentheses.

Question 61

Using the graph below, find the given limit.

- $\lim _ { x \rightarrow \infty }$ (2f(x) - 1)

A) 6
B) 0
C) 5
D) 2

Question 62

Calculate the following limit(s) if they exist.
Enter just an integer or a fraction.

Question 63

Does (where a ≠ 0) exist?
Enter your answer as either "yes" or "no" or "does not exist".

Question 64

Calculate the following limit(s) if they exist.
where Enter your answer as a fraction, integer, or the words "does not exist".

Question 65

For the following function g(x), determine whether or not $\lim _ { x \rightarrow 2 }$ g(x) exists. If so, give the limit.

A) 2
B) -6
C) 0
D) does not exist

Question 66

If f(t) = $\frac { 1 } { x ^ { 2 / 3 } }$ , then $\lim _ { h \rightarrow 0 }$ $\frac { f ( - 8 + h ) - f ( - 8 ) } { h }$ equals

A) f'(8)
B) $\frac { 1 } { 48 }$
C) f'(0)
D) none of these

Question 67

For the following function f(x), determine whether or not $\lim _ { x \rightarrow 0 }$ f(x) exists. If so, give the limit.

A) 0
B) - $\frac { 1 } { 3 }$
C) 1
D) does not exist

Question 68

Calculate the following limit(s) if they exist.
Enter just an integer or a fraction.

Question 69

Calculate the following limit(s) if they exist.

- $\lim _ { x \rightarrow \infty }$ $\frac { 2 x ^ { 2 } + 1 } { x ^ { 2 } + 1 }$

A) 0
B) 2
C) ?
D) 1

Question 70

Determine whether or not the following limit exists. If, so compute the limit. $\lim _ { x \rightarrow - 1 }$ $\frac { x ^ { 2 } - 1 } { x + 1 }$

A) -2
B) 1
C) does not exist
D) 0
E) none of these

Question 71

Calculate the following limit(s) if they exist.
+ 1
Enter either a fraction, integer, or the words "does not exist"

Question 72

Using the graph below, find the given limit.

- $\lim _ { x \rightarrow 0 }$ $[ \mathrm { f(x) }]^2$

A) 4
B) 9
C) 0
D) 6

Question 73

Calculate the following limit(s) if they exist.
+ 1
Enter either a fraction in lowest terms, an integer, or "does not exist".

Question 74

Calculate the following limit(s) if they exist.
Enter either a fraction, integer, or "does not exist".

Question 75

Determine whether or not the following limit exists. If, so compute the limit. $\lim _ { x \rightarrow 3 }$ $\frac { \sqrt { x } - 4 } { x ^ { 3 } + 27 }$

A) does not exist
B) 0
C) $\frac { \sqrt { 3 } - 4 } { 54 }$
D) $\sqrt { 3 }$ - 4
E) none of these

Question 76

Calculate the following limit(s) if they exist.
where Enter your answer as an integer, fraction in lowest terms, or the words " does not exist"

Question 77

Calculate the following limit(s) if they exist.
Enter either a fraction, integer, or the words "does not exist".

Question 78

For the following function g(x), determine whether or not $lim _ { x \rightarrow 2 }$ g(x) exists. If so, give the limit.

A) 3
B) 0
C) 1
D) does not exist

Question 79

For the following function g(x), determine whether or not $\lim _ { x \rightarrow - 1 }$ g(x) exists. If so, give the limit.

A) 2
B) 0
C) 1
D) does not exist

Question 80

Let f(x) = $\frac { x ^ { 3 } - 9 x } { 2 x + 6 }$ . Does $\lim _ { x \rightarrow - 3 }$ f(x) exist?