MATH 180 Study Guide - Final Guide: Quotient Rule

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turquoisegnu128

13 Dec 2018

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Problem 1 solution: calculate each limit below. (a) lim x 7(cid:18) 14. 2 x 7(cid:19) x2 7x . Solution: (a) the least common denominator of the function is x2 7x. Thus, the function can be written as follows: f (x) = Therefore, the limit of f (x) as x 7 is x 7 (cid:18) . 2 x 7(cid:19) = lim x2 7x x(cid:19) = . 7 (b) the function is rational and the degrees of the numerator and denominator are the same. Therefore, the limit of f as x is the ratio of the leading coe cients. lim x . 3: if f (x) = 3x + 1, calculate. Problem 2 solution lim h 0 f (x + h) f (x) h. Solution: it is easiest to calculate the limit by recognizing that, by de nition, Given that f (x) = 2x + 1, we can use the chain rule: lim h 0 f (x + h) f (x) h.

Related Questions

1. Use the definition of continuity and the properties of limitsto show that the function is continuous at the given number(a).

f(x) = (x+3x^5)^4 , a= (-1)

lim f(x) as x approaches -1 (x+3x^5)^4

= (lim x +3 times lim x^5)^4

= (-1 +3 (???)^5)^4 ------got stuck here

= ???

f(-1) = ??

2. Find the average value f(ave) of the function f on the giveninterval

f(x)= 6x -x^2 , [0, 4]

3. Find the area of the shaded region between a curve and aline:

y = 9x-x^2 and y = 2x

the functions intersect at (0,0) and (7, 14)

I need to figure out the area of the upper half, above the y =2xline

it is a concave down parabola

4. (21 pts) (I) (6 pts) Determine the value of a constant b for which the function g is continuous at 0. Show your work. Justify your answer! 2 x +b if x<0, X-5 g(x) = if x 20, x#4. We have to show that lim g (x) = g (0). g(0) = 0 +18 = -1; lim X0* 0 + 16 ) = lim x + 16 **0* x2 - 160 - 16- Since lim g (x) has to exist (in order for g to be continuous at x = 0), it has to be true that x-0- x0+ X- 0 lim g(x) - lim g(x) = lim g(x). Therefore, -1, and , so b = 5. so, if b = 5, lim g (x) = g(0), and therefore, g is continuous at x = 0. (II) (4 pts) Evaluate the limit. Show your work! 2. b limx- (a) lim g(x) = lim *+- 2 x + b X - 5 : 1 * +16 10 g (x) = lim x++ - Elim x2 - 16 x++ (x - 10). OR 2x+5. 2; (a) lim g (x) = lim x--00 X- (b) lim x++ 2 + 16 g(x) = lim **** x2 - 16 X-+ (III) (4 pts) Determine whether g has any horizontal asymptotes. If so, write the equation (or equations) of all horizontal asymptotes. y = 2, by part (II) (a) y = 0, by part (II) (b) (IV) (7 pts) Determine whether g has any vertical asymptotes. If so, write the equation (or equations) of all vertical asymptotes. Show your work. Justify your answer! 5 + 16 There is no vertical asymptote at x = 5, because lim g(x) = x+5 52-16' -8 + b (x) = there is no vertical asymptote at x = -4, because lim X -4 - 4 - 5 But, lim g(x) = lim + 16 x+4+ x2 - 16 lim_ (x + 16) -20 x+4+ (x - 4) 60 (x + 4) +8 (or the numerator goes to 20, and the denominator is positive and goes to o, as x + 4*) = +00 So, the function g has only one vertical asymptote, x = 4.

Find the derivative of the function and evaluate thederivative at the given x-value.

f(x) = 2x² at x = 1

A) f' (x) = 4x; f' (1) = 2
B) f ' (x) = 2x; f ' (1) = 2
C) f' (x) = 4x²; f' (1) = 4
D) f' (x) = 4x; f' (1) = 4

f(x) = 6x + 2; [-1, 2]

A) Absolute maximum: 12, absolute minimum: -6
B) Absolute maximum: -1, absolute minimum: 2
C) There are no absolute extrema.
D) Absolute maximum: 14, absolute minimum: -4

f(x) = (-5x + 7)⁴

A) f '(x) = -5(-5x + 7)³
B) f '(x) = -20(-5x + 7)⁴
C) f '(x) = 4(-5x + 7)³
D) f '(x) = -20(-5x + 7)³

A)
B)
C)
D)

A company estimates that the daily revenue (in dollars) from thesale of x cookies is given by

R(x) = 885 + 0.02x + 0.0003x²

Currently, the company sells 900 cookies perday.

Use marginal revenue to estimate the increase in revenueif the company increases sales by one cookie per day.

A) $92.00
B) $0.56
C) $56.00
D) $0.92

f(x) = -3 - 7x; [-3, 1]

A) Absolute maximum: -10, absolute minimum: -24
B) Absolute maximum: 18, absolute minimum: -10
C) There are no absolute extrema
D) Absolute maximum: 24, absolute minimum: -4

A)
B)
C)
D)

s(x) = -x² - 20x - 19

A) Relative maximum at ( -20, -19)
B) Relative maximum at ( 10, 81)
C) Relative maximum at ( -10, 81)
D) Relative minimum at ( 20, -19)

Differentiate.

f(x) = (5x + 4)²

A) f'(x) = 10(5x + 4)²
B) f'(x) = 5(5x + 4)
C) f'(x) = 2(5x + 4)
D) f'(x) = 10(5x + 4)

10.

Find the absolute maximum and absolute minimum values ofthe function, if they exist, over the indicated interval. When nointerval is specified, use the real line (-?, ?).

f(x) = -21; [ -7, 7]

A) Absolute maximum: 21, absolute minimum: 0
B) Absolute maximum: 21, absolute minimum: -21
C) Absolute maximum: -21, absolute minimum: -21
D) There are no absolute extrema.

11.

A)
B)
C)
D)

12.

f(x) = -6x² - 2x - 7

A) Relative maximum at
B) Relative maximum at
C) Relative maximum at
D) Relative maximum at

13.

f(x) = 0.2x² - 2.4x + 5.9

A) Relative minimum at ( 6, -1.3)
B) Relative maximum at ( 6, -1.3)
C) Relative minimum at ( -6, 27.5)
D) Relative minimum at ( 6, 0)

14.

A grocery store estimates that the weekly profit (in dollars)from the production and sale of x cases of soup is given by