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Question : Find the derivative. 1) y = (4x + 3)^5 : 2151678

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Find the derivative.

1) y = (4x + 3)^{5}

A) (dy/dx) = 5(4x + 3)^{4}

B) (dy/dx) = 4(4x + 3)^{4}

C) (dy/dx) = (4x + 3)^{4}

D) (dy/dx) = 20(4x + 3)^{4}

2) y = √(4x + 2)

A) (dy/dx) = (1/√(4x + 2))

B) (dy/dx) = (2/√(4x + 2) )

C) (dy/dx) = (8/√(4x + 2))

D) (dy/dx) = (4/√(4x + 2))

3) f(x) = (x^{3} - 8)^{2/3}

A) f'(x) = (x/(3)√(x^{3} - 8)

B) f'(x) = (2x^{2}/(3)√(x^{3} - 8)

C) f'(x) = (x^{2}/(3)√(x^{3} - 8)

D) f'(x) = (2x/(3)√(x^{3} - 8)

4) y = (x^{-2} + x)^{-3}

A) (dy/dx) = (3x^{4}(2 - x^{3})/(1 + x^{3})^{4})

B) (dy/dx) = (3x^{5}(2 - x^{3})/(1 + x^{3})^{4})

C) (dy/dx) = (3x^{5}(2 - x^{3})/(1 + x^{3})^{3})

D) (dy/dx) = (3x^{4}(2 - x^{3})/(1 + x^{3})^{3})

5) y = (x + 1)^{2}(x^{2} + 1)^{-3}

A) (dy/dx) = 2(x + 1)(x^{2} + 1)^{-4}(2x^{2} - 3x - 1)

B) (dy/dx) = 2(x + 1)(x^{2} + 1)^{-4}(2x^{2} + 3x - 1)

C) (dy/dx) = -2(x + 1)(x^{2} + 1)^{-4}(2x^{2} - 3x - 1)

D) (dy/dx) = -2(x + 1)(x^{2} + 1)^{-4}(2x^{2} + 3x - 1)

6) f(x) = (5/(2x - 3)^{4})

A) f'(x) = (-40/(2x - 3)^{5})

B) f'(x) = (5/8(2x - 3)^{5})

C) f'(x) = (-40/(2x - 3)^{3})

D) f'(x) = (5/4(2x - 3)^{3})

7) y = (2x - 1)^{3}(x + 7)^{-3}

A) (dy/dx) = 45(2x - 1)^{2}(x + 7)^{-3}

B) (dy/dx) = 45(2x - 1)^{2}(x + 7)^{-4}

C) (dy/dx) = 45(2x - 1)^{3}(x + 7)^{-4}

D) (dy/dx) = 45(2x - 1)^{3}(x + 7)^{-2}

8) y = x √(x^{2} + 1)

A) (dy/dx) = (x^{2} + 1/√(x^{2} + 1))

B) (dy/dx) = (√(x^{2} + 1)/x^{2} + 1)

C) (dy/dx) = (2x^{2} + 1/√(x^{2} + 1))

D) (dy/dx) = (√(x^{2} + 1)/2x^{2} + 1)

9) y = ((3)√(x^{2} + 3)/x)

A) (dy/dx)= (-x^{2} - 9/3x^{2}(x^{2} + 3)^{2/3})

B) (dy/dx)= (3/x^{2}(x^{2} + 3)^{2/3})

C) (dy/dx)= (x^{2} + 9/3x^{2}(x^{2} + 3)^{2/3})

D) (dy/dx)= (-3/x^{2}(x^{2} + 3)^{2/3})

10) y = (3x^{2} + 5x + 1)^{3/2}

A) (dy/dx) = (3/2)(6x + 5)(3x^{2} + 5x + 1)^{1/2}

B) (dy/dx) = (3x^{2} + 5x + 1)^{1/2}

C) (dy/dx) = (6x + 5)(3x^{2} + 5x + 1)^{1/2}

D) (dy/dx) = (3/2)(3x^{2} + 5x + 1)^{1/2}

The table lists the values of the functions f and g and their derivatives at several points. Use the table to find the indicated derivative.

11)

Find D_{x}(f[g(x)]) at x = 3.

A) 3

B) -5

C) -15

D) 1

12)

Find D_{x}(g[f(x)]) at x = 2.

A) -9

B) 2

C) -12

D) -6

Find the equation of the tangent line to the graph of the given function at the given value of x.

13) f(x) = (x^{2} + 12)^{3/4}; x = 2

A) y = (3/2)x

B) y = (3/4)x + 5

C) y = (3/2)x + 5

D) y = (3/2)x + 11

14) f(x) = x^{3}√(x^{3} + 15); x = 1

A) y = (99/8)x - (67/8)

B) y = (143/12)x - (37/3)

C) y = (99/8)x - (61/8)

D) y = (143/12)x + (37/3)

Find all values of x for the given function where the tangent line is horizontal.

15) f(x) = √(x^{2} + 12x + 48)

A) -6

B) 0, -6

C) 0, 6

D) -6, 6

16) f(x) = (x/(x^{2} + 3)^{3})

A) 0

B) 0, ± (√(15)/5)

C) ± (√(3)/5)

D) ± (√(15)/5)

Solve the problem.

17) The total revenue from the sale of x stereos is given by R(x) = 2000(1 - (x/800))^{2}. Find the average revenue from the sale of x stereos.

A) (2000/x)(1 - (x/800))^{2}

B) 2000x(1 - (x/800))^{2}

C) (1000/x)(1 - (x/800))^{2}

D) 1000x(1 - (x/800))^{2}

18) The total revenue from the sale of x stereos is given by R(x) = 2000(1 - (x/700))^{2}. Find the marginal average revenue.

A) 2.86 - (700/x^{2})

B) 2.86 - (2000/x^{2})

C) 0.004 - (700/x^{2})

D) 0.004 - (2000/x^{2})

19) $2600 is deposited in an account with an interest rate of r% per year, compounded monthly. At the end of 8 years, the balance in the account is given by A = 2600(1 + (r/1200))_{96}. Find the rate of change of A with respect to r when r = 10.

A) (dA/dr) = 211.48

B) (dA/dr) = 209.73

C) (dA/dr) = 461.38

D) (dA/dr) = 457.57

20) The formula E = 1000(100 - T) + 580(100 - T)^{2} is used to approximate the elevation (in meters) above sea level at which water boils at a temperature of T (in degrees Celsius). Find the rate of change of E with respect to T for a temperature of 67°C.

A) 39,280 m/°C

B) -39,280 m/°C

C) -78,140 m/°C

D) -38,280 m/°C

21) A circular oil slick spreads so that as its radius changes, its area changes. Both the radius r and the area A change with respect to time. If dr/dt is found to be 1.8 m/hr, find dA/dt when r = 27.4 m.

A) 98.64π m^{2}/hr

B) 197.28π m^{2}/hr

C) 49.32π m^{2}/hr

D) 24.66π m^{2}/hr

22) When an amount of heat Q (in kcal) is added to a unit mass (in kg) of a substance, the temperature rises by an amount T (in degrees Celsius). The quantity dQ/dT, called the specific heat, is 0.18 for glass. If dQ/dt = 10.8 kcal/min for a 1 kg sample of glass at 20.0°C, find dT/dt for this same sample.

A) 1.944 kcal/min

B) 63 kcal/min

C) 60 kcal/min

D) 29 kcal/min

23) Suppose that a demand function is given by q = D(p) = 21(7 - (p^{2}/√(p^{3} + 1))) ,

where q is the demand for a product and p is the price per unit in dollars. Find the rate of change in the demand for the product per unit change in price.

A) (dq/dp) = (-21p^{4} + 84p/(p^{3} + 1)^{1/2})

B) (dq/dp) = (-21p^{4} - 84p/2(p^{3} + 1)^{3/2} )

C) (dq/dp) = (21p^{4} - 84p/2(p^{3} + 1)^{3/2} )

D) (dq/dp) = (-21p^{4} - 84p/2(p^{3} + 1)^{1/2} )

24) The concentration of a certain drug in the bloodstream t minutes after swallowing a pill containing the drug can be approximated using the equation C(t) = (1/8)(3t + 1)^{-1/2}, where C(t) is the concentration in arbitrary units and t is in minutes. Find the rate of change of concentration with respect to time at t = 8 minutes.

A) - (1/40) units/min

B) - (3/2000) units/min

C) - (1/2000) units/min

D) - (3/80) units/min

Provide an appropriate response.

25) What rule is applied first to find the derivative of the function

f(x) = (4x^{5} + 6)√(((4x + 2)^{5}/2x^{4} - 2))?

A) Product rule

B) Quotient rule

C) Power rule

D) Square root rule

26) What rule is applied first to find the derivative of the function

f(x) = √(((3x - 5)^{2}(x^{3} + 6)/3x - 6))?

A) Product rule

B) Quotient rule

C) Square root rule

D) Power rule

27) What rule is applied first to find the derivative of the function

f(x) = ((2x^{5} + 2)√(5x^{3} + 2)/5x - 6)?

A) Square root rule

B) Product rule

C) Power rule

D) Quotient rule

28) What rule is applied first to find the derivative of the function

f(x) = (((2x^{4} + 1)√(4x^{3} - 2)/5x + 5))^{3}

A) Power rule

B) Quotient rule

C) Product rule

D) Square root rule

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

29) How is the graph of y = f(x) = x^{5} + 4x + 1 related to the graph of y = g(x) = (x - 4)^{5} + 4(x - 4) + 1? How is the slope of the graph of g(x) at x = a related to the slope of the graph of f(x) at x = a - 4?

30) How is the graph of y = f(x) = x^{2} - 2x + 1 related to the graph of y = g(x) = (3x)^{2} - 2(3x) + 1? How is the slope of the graph of g(x) at x = a related to the slope of the graph of f(x) at x = a?

31) Find the derivative of f(x) = (5x + 3)^{3} in two ways. First multiply out and differentiate. Then use the power rule. Show that the answers are equivalent.

32) Use the chain rule to prove the quotient rule for f(x) = (g(x)/h(x)).