Calc final for Mrs. Black

(3.9) The function ƒ is defined by f(x) = x / (x+2). What points (x,y) on the graph of ƒ have the property that the line tangent to ƒ at (x,y) has a slope 1/2?

(0,0) and (-4,2)

(2.9)- If x^2+xy+y^3=0, then, in terms of x and y, dy/dx=

- (2x+y / x+ 3y^2)

(2.3) d/dx cos^2(x^3)=

-6x^2sin(x^3)cox(x^3)

(#1.2) lim as x➞∞ 7x³-2x²+3x/-x³-2x+7

-7

(2.15)- A particle moving along the x-axis such that at any time t>0 its position is given by x(t)= 2-30t+81t^2-20t^3. For what values of t is the particle moving to the left?

0 <t < 1/5, t>5/2

(2.18)- lim h-->0... ln(4+h)-ln(4)/ h is

1/4

(3.3) If f(x) = (x^2 -2x -1) ^2/3, then f'(0) is

4/3

(#1.7) if the graph of y=ax+b/x+c has a horizontal asymptote y=2 and a vertical asymptote x=-3, then a+c=

5

(3.18) Let F(x) = { (x^2 + x) / x, x≠0 _______{ 1 , x = 0 Which of the following statements are true of F? I. F is defined at x = 0. II. lim(x->0) F(x) exists. III. F is continuous at x=0

I only

(3.17) Which of the following functions are continuous for all real numbers x? I. y= x^4/3 II. y= 3^√(3x -1) III. y = (3x -1) / (4x^2 +5)

I, II, and III

(3.13) Find the derivative of the algebraic function. R(s) = (s^5 + 5)^4

R'(s)= 20s^4•(s^5 + 5)^3

(2.2) At x=3, the function given by f(x)={x^2, x<3 6x-9, x>3 is...

both continuous and differentiable

(2.4) f(x)= xln(x^2), then f'(x)=

ln(x^2)+2

(3.15) Find an equation of the tangent line to the graph of ƒ at the given point. f(x) = (s -4)(s^2 -3), at (2, -2)

y = -7s + 12

(3.14) Find an equation to the tangent line for the graph of ƒ at the given point. f(x) = (5x^4 + 4)^2, at (1, 81)

y = 360x - 279

(#1.8) the line y=5 is a horizontal asymptote to the graph of which of the following functions?

y=20x²-x/1+4x²

(1.11)- Find the value of k such that the function f(x)= {kx+7, x<2 .... 3x^2-8, x>2} for all values of x.

-3/2

(2.6) If (x+2y) dy/dx = 2x-y, what is the values of d^2y/dy^d at the point (3,0)?

-10/3

(#1.4) find the limit: lim as x➞-7 x²+11x+28/x+7

-3

(3.1) If f(x) = -x^3 + x + 1/x, then f'(-1) =

-3

(3.11) Let ƒ and "g" be differentiable functions such that f(1)=2, f'(1)=3, f'(2)=-4, g(1)=2, g'(1)=-3, g'(2)=5. If h(x) = f(g(x)), then h'(1) =

12

(3.2) An equation of the line tangent to the graph of f(x) = (2x+3/(3x-2) at the (1,5) is

13x + y =18

(3.10) What is the instantaneous rate of change at x=2 of the function ƒ given by f(x) = (x^2 -2) / (x-1)?

2

(2.5) If y= (x^3-cosx)^5, then y'=

5(x^3-cosx)^4 times (3x^2+sinx)

(2.17)- If f(x) = 7x-3 + lnx, then f'(1)=

8

(1.14)- Which of the following correctly describes the discontinuities associated with f(x)= x^2-2x-3 / x^2-9

A hole at x=3, a vertical asymptote at x= -3

(3.16) Find the derivative of the algebraic function.n Q(t) = v( 9- 5/(v + 7))

Q'(v) = (406 + 126v + 9v^2) / (v + 7)^2

(2.11)- *insert image* lim x-->a ... f(x) does not exist for which of the following values of a?

a= 1, 3 only

(3.12) f(x)= √x•(2-x^3)

f'(x) = -3x^2.5 + (2-x^3) / (2√x)

(3.5) f(x) = 2x(x^2 + 1)^4

f'(x) = 2(x^8 + 4x^6 + 6x^4 +4x^2 + 11) + 2(8x^7 + 24x^5 + 25x^3 + 8x)

(3.7) Find the derivative and simplify: f(x) = 2/3x^5 - 3/5x^3 +1/2x

f'(x) = 3.33x^4 -1.8x^2 + 1/2

(1.16)- Let f be a continuous function on the interval [-3,6]. If f(-3)= -1, and f(6)=3, then the Intermediate Value Theorem guarantees that...

f(c)=1 for at least on c between -3 and 6

(1.13)- Determine the function whose graph has vertical asymptotes at x= +-2 and a horizontal asymptote at y=0.

f(x)= 3x / x^2-4

(2.14)- A particle moves along the x-axis with its position at time, t, given by x(t)=(t-a)(t-b), where a and b are constants and a does not equal b. For which of the following values of t is the particle at rest?

t= a+b / 2

(#1.5) find the limit: lim as x➞1 (2- (5/(x-1)²))

-∞

(1.15)- Find the limit: lim (x--> infinity)... sqr root(4x^2-1 / x^2)

0

(#1.1) find the limit: lim as x➞3+ √2x-5

1

(2.1) A particle moves along the X-axis so that at any time t is given by x(t)=te^-2t. For what values of t is the particle at rest?

1/2 only

(#1.9) if a≠0, then lim as x➞a x²-a²/x⁴-a⁴ is

1/2a²

(2.7) Let f(x)=(2x+1)^3 and let g be the inverse function of f. Given that f(0) = 1, what is the value of g'(1)?

1/6

(3.8) What is the instantaneous rate of change for f(x) = (x^3 + 3x^2 + 3x +1) / (x+1) at x=2?

6

(3.4) If f(x) = √(x^2-4) and g(x) = 3x-2, then the derivative of f(g(x)) at x=3 is

7/√5

(2.16)- At which points is the tangent line to the curve of 8x^2+2y^2=6xy+14 vertical? I. (-2,-3) II. (3,8) III. (4,6)

I and III only

(1.10)- Let g be a continuous function on the closed interval [0,1]. Let g(0)=1 and g(1)=0. Which of the following is not necessarily true

There exists a number h in [0,1] such that g(h)=3/2

(1.17)- Let f be a function that is continuous on the closed interval [2,4] with f(2)=10 and f(4)=20. Which of the following is guaranteed by Intermediate Value Theorem?

f(x)=13 has at least one solution in the open interval (2,4)

(#1.3) the graph of the f is shown above. Which of the following statements is false?

lim as x➞4 f(x) exists

(2.13)- If y= xsinx, then dy/dx=

sinx + xcosx

(2.10)- *insert image* f(x) is discontinuous for

x= 1, 2. 3, 4 only

(2.12)- *insert image* f(x) is not differentiable at

x= 1, 3 only

(#1.6) find all vertical asymptote(s) of f(x)=x-2/x²-4

x=-2

(1.12)- Which of the following functions has a horizontal asymptote at y= -1/2?

x^3 / 1-2x^3

(2.8) If y=x^2e^x, then dy/dx

xe^x(x+2)