CALCULUS EXAM

2sin(t)

A body moving in simple harmonic motion is described by the function s = 2cos(t). Find the jerk.

A) 400 ft B) t = 8 t=2 C) -32ft/s^2 D) 10 s

A dynamite blast propels a rock with a velocity of 160 ft/sec. It reaches a height of s = 160t - 16t^2 ft after t seconds. A) How high does the rock go? B)What is the velocity of the rock 256 ft above the ground on the way up? Down? C) What is the acceleration of the rock? D) When does the rock hit the ground?

A) - 4 B) 0 m/s C) 4 m/s D) 2 m/s^2 E) changes direction at 2 seconds; t= 2

A particle moves so that t >_ 0 , given by the function s(t) = t^2 - 4t + 3. S is measured in meters, t is in seconds. A) Find displacement during first two seconds B) Find average velocity in first four seconds C) Find instantaneous velocity at t = 4 D) Find acceleration at t = 4 E) Describe the motion

71 m/s

A projectile is shot up from the surface of the earth and reaches a height of s = -4.9t^2 + 120t meters after t seconds. Find the velocity of the projectile after 5 seconds.

^y/^t = f(t + h) - f(t)/(t + h) - t = f(2) - f(0)/2-0 = 16(2)^2 - 16(0)^2/2 32 ft/s

A rock breaks loose from the top of a tall cliff. What is its average speed during the first 2 seconds of fall? f(t) = 16t^2

A) 2πr B)20π C)square inches of area per inch of radius

A) Find the rate of change of area A with respect to its radius r B) Evaluate the rate of change of A at r = 5 and at r = 10 C) if r is measured in inches and area is measured in square inches, what unit of measurement would be appropriate for dA/dr

Second derivative from speed, then velocity

Acceleration is what form of derivative?

a) N'(t) = 0.6t^2 + 20t + 160 b) 486

An internet service provider has modelled its project number of business subscribers by B(t) = 50t + 2500 and the projected average number of internet connections per business subscriber as C(t) = 0.004t^2 + 3.2 where "t" is time in weeks, over the past year. The projected total number of business internet connections is then N(t) = B(t) x C(t) a) determine the growth rate of the number of business connections after t weeks b) 12 weeks

lim/x->0 f(a+h) - f(a)/h Slope = -2 Equation of tangent : y = -2x - 1 Equation of normal : y = 1/2x - 7/2

Consider the function f(x) = x^2 - 4x. Find the slope of the tangent to the curve at the point x = 1. Find the equations of the tangent and normal lines.

A line passing through the points on a curve (avg rate of change)

Define a secant line to a curve.

A line that touches the curve (measure of instant rate of change)

Define a tangent line to a curve.

= 1

Determine the average rate of change for f(x) = 2x^2 - 3x + 7 over the interval [-2, 4]

y' = 1 - 2cos(x)/(cos(x)-2)^2

Determine the derivative of y = sin(x)/cos(x) - 2

1. No HA, doesn't settle to a particular value. f(x) -> infinity 2. f(x) -> 0 H.A. @ y=0 3. f(x) -> 2 H.A. @ y=2

Determine the horizontal asymptote for the following : 1. f(x) = 2x^5 + x^4 - x^2 + 1/3x^2 - 5x + 7 2. f(x) = 4x^2 - 3x + 5/2x^3 - 3x - 2 3. f(x) = 4x^2 - 3x + 5/2x^2 - 3x - 2

Slope = 45/22 Equation : y - 9_/3/32 = 45/22(x - π/3)

Determine the slope and equation of a tangent line to the curve y = sin^5(x) where x = π/3

V.A. @ x = 3 H.A. @ n=m @ y=-1

Determine the vertical and horizontal asymptotes for the function : f(x) = x/3-x

f(x) = (x - 3)(x^2 + 3x + 2)/(x+3)(x-3) =x^2 + 3x + 2/x+3 Numerator factoring : by decomposition (x-3 is a factor) Denominator factoring : (x+3)(x-3) Conclusion : x=3 there is a point of discontinuity x=-3 there is an asymptotic discontinuity lim/x->3 x^3 - 7x -6/x^2 - 9 =10/3 g(x) {x^3 - 7x - 6/x^2 -9, x cannot = 3 AND {10/3, x = 3

Determine where the function is discontinuous. State the nature of the discontinuities. f(x) = x^3 - 7x - 6/x^2 - 9

1. 4x - 3 2. 15x^2 - 1/2_/x

Differentiate : 1. f(x) = 2x^2 - 3x +4 2. g(x) = 5x^3 - _/x

A(x) = x^2 - 3x + 2/ x^2 + 4x + 1 = 1/13

Differentiate and state the slope of the tangent to the curve at x = 2.

-12x^3sin(3x^4 - 2)

Differentiate cos(3x^4 - 2)

f'(a) = 1/2_/a; (0, infinity)

Differentiate the function f(x) = _/x at x=a

f'(5) = 8

Differentiate the function f(x) = x^2 - 2x at x = 5

1. 14x^6 2. -2/x^2 3. 4u^7

Differentiate using the constant multiple rule : 1. f(x) = 2x^7 2. f(x) = 2/x 3. d/du(1/2u^8)

1. f'(x) = 0 2. f'(x) = 0

Differentiate using the constant rule : 1. f(x) = 6 2. f(x) = 280000

1. f'(x) = 7x^6 2. f'(x) =20x^19 3. y' = 4t^3 4. 8u^7 5. f'(x) = -3/x^4 6. f'(x) = 1/2_/x

Differentiate using the power rule : 1. f(x) = x^7 2. f(x) = x^20 3. y = t^4 4. d/du(u^8) 5. f(x) = 1/x^3 6. f(x) = _/x

y' = 5_/x^3 + 6_/x - 3/_/x

Differentiate using the product rule : y = _/x(2x^2 + 4x - 6)

H.A. @ y=0 f(10) = 8/4-(10)^2 = -0.08 V.A. @ x=+-2 Nature of the function around x=-2 left x->-2- = neg right x->-2+ =pos Nature of the function around x=2 left x->2- = pos right x->2+ =neg

Find all the asymptotes and describe the end behavior of the function to the left and right of the vertical asymptote(s) for : f(x) = 8/4-x^2

y'' = 20x^3

Find d^2y/dx^2 if y = x^5

f'(2) = -1/16

Find dy/dx for f(x) = 1/x+2 at x=2

f'(x) = 1/2_/(x+1)

Find dy/dx for f(x) = _/(x+1)

y = -1/18x + 5/2

Find the equation of the tangent line to a curve of the function y = _/x + 3/_/x at the point (9,2).

lim/x->+infinity POS : f(5) = 0.923076923 f(10) = 0.980198019 f(15) = 0.991150442 = 1 lim/x->-infinity NEG: f(5) = 0.923076923 f(10) = 0.980198019 f(15) = 0.991150442 = 1

Find the horizontal asymptote(s) for : lim/x->infinity (x^2 - 1/x^2 + 1)

g'(3) = -25 g''(3) = -16

Find the slope of the tangent and the rate of change of the slope of the tangent at x = 3. g(x) = 2 - 4x + x^2 - x^3

^y/^t = 16(2)^2 - 16(2)^2/2-2 = 64 - 64/2 - 2 = 0/0 *first theoretical limit* ^y/^t = 16(2+h)^2 - 16(2)^2/h = 16(4 + 4h +h^2) - 64/h = 64 + 64h + 16h^2 - 64/h = 64h + 16h^2/h = h(64 + 16h)/h = 64 + 16h where h trends to a value of 0 = 64 ft/s

Find the speed of a falling rock at the instant t=2 if f(t) = 16t^2.

2xcos(x^2)/3 3^_/sin(x^2)^2

Find y' for y = 3^_/sin(x^2)

y' = 2xsin(x) + x^2cos(x)

Find y' for y = x^2sin(x)

cos(x)/_/2sin(x) - 2

Find y' given y = _/2sin(x) - 2

y'' = tan^2(x)sec(x) + sec^3(x)

Find y'' for y = sec(x)

y = 3/(x-1)(x-1) NPV's : x cannot = 1 NO point of discontinuity NO jump discontinuity Infinite asymptotic discontinuity at x = 1 Domain : (-infinity,1)(1, infinity) Lim x->1- (left) : +infinity Lim x->1+ (right) : +infinity

For the following function, identify and classify any points of discontinuity. State the domain of the function and the limit if it exists. y = 3/(x-1)^2

By getting a result of 0 that results in division by 0 (undefined - NPV's). Ex : 1/x+2 has a vertical asymptote @ x=-2

How do we get a vertical asymptote?

False

If a function is to be continuous at a given point, then the limit, at that point does not exist. True or False?

48(4x -2)^3

If f(x) = 3(4x -2)^2 find the derivative

1. lim/x->0 sin(x) = 0 2. lim/x->0 cos(x) = 1 3. lim/x->0 sin(x)/x = 1

Label *PICTURE 2*

1. Removable (point) discontinuity : where a function on the top of a division cancels an identical function on the bottom. On a graph this is an open white dot Ex : x^2 - 7x + 12/(x-4) =(x-3)(x-4)/(x-4) 2. Jump discontinuity : piecewise function with left and right side limits that must agree for the limit to exist. On graph this causes disconnected lines Ex : f(x) { x^2 + 2 ; x<2 and x-5 ; x>_ 2 3. Infinite (Asymptotic) discontinuity : division by 0. On graph has a clear asymptote separating two sides to a graph Ex : f(x) = 9/x^2 OR lim/x->0 (9/x^2) = infinity 4. Oscillating discontinuity : two functions together, usually with a trig function. On a graph this looks like wave frequency Ex : f(x) = sin(1/x)

List and describe the 4 types of discontinuity.

1. If the leading power on the bottom is higher than the leading power on the top, lim/x->infinity = f(x) = 0 2. If the leading power on the bottom is equal to the leading power on the top, lim/x->infinity = f(x) = a/b (top coefficient divided by bottom coefficient) 3. If the leading power on the bottom is less than the leading power on the top, lim/x->infinity = f(x) = infinity

Make three general statements about the end behavior of horizontal asymptotes.

Change in y / change in t ---> distance / time

START OF 2.1 The average rate of change aka. average speed of an object is defined as ___.

+infinity : f(1) = 1 f(10) = 0.1 f(100) = 0.01 f(1000) = 0.001 ... f(25000) = 0.00004 -infinity : f(-1) = -1 f(-10) = -0.1 f(-100) = -0.01 f(-1000) = -0.001 ... f(-25000) = -0.00004 trending towards 0

START OF 2.2 Consider : f(x) = 1/x as x->+-infinity

1. f(c) = 2 lim/x->c f(x) = 2 lim/x->c f(x) = f(c) YES 2. f(c) = 1 lim/x->c f(x) = DNE lim/x->c f(x) = f(c) NO 3. f(c) = 2 lim/x->c f(x) = 1 lim/x->c f(x) = f(c) NO

START OF 2.3 Using PICTURE 3, determine if y = f(x) is continuous at : 1. x=3 2. x=1 3. x=2

f(x2) - f(x1)/x2-x1 = [(3)^3 - 3] - [(1)^3 - 1]/2 =12

START OF 2.4 Determine the average rate of change for f(x) = x^3 - x over the interval [1,3].

f'(x) = 3x^2

START OF 3.1 Differentiate the function f(x) = x^3.

1. At a corner, where one sided derivatives differ. On a graph this is at the tip of a V-shaped graph. Ex : f(x) = |x| One side has a slope of -1 and one has a slope of 1 2. At a cusp, where the slopes of the secant line approach infinity from one side and negative infinity from the other. On a graph this is a bent, curvy V-shaped graph. Ex : f(x) = x^2/3 3. At a vertical tangeant, where the slopes of the secant lines approach either infinity or -infinity. On a graph this is an S-shaped curve . Ex : f(x) = 3^_/x 4. At a discontinuity. On a graph this looks like two separated lines. Ex : f(x) = {x = -1; x < 0 {x = 1; x>_ 0

START OF 3.2 List how f'(a) might fail to exist.

f'(x) = 3x^2 - 6x + 5

START OF 3.3 Use the definition of a derivative to determine the derivative of the function f(x) = x^3 - 3x^2 + 5x + 2

No derivative

START OF 3.4 Speed is what form of derivative?

PICTURE 4

START OF 3.5 KNOW TRIG DERIVATIVES

12x(3x^2 + 2)

START OF 4.1 If f(x) = (3x^2 +2)^2 find the derivative

Left : 0 Right : 2 Since our "sided" derivatives do not agree, we don't have an overall derivative at zero

Show that the following function has left hand and right hand derivatives at x=0, but no derivatives there. y= {x^2; x _< 0 {2x; x > 0

lim/x->2 (x^2 +2x +4/x+2) = 2^2 + 2(2) + 4/2+2 = 3

Solve : lim/x->2 (x^2 +2x +4/x+2)

x^3 + 4x^2 - 3 = (2)^3 + 4(2)^2 - 3 = 21

Solve : lim/x->2 (x^3 + 4x^2 - 3)

lim/x->3 [x^2 (2-x)] = [3^2 (2-3)] = -9

Solve : lim/x->3 [x^2 (2-x)]

1. lim/x->0 sin(x)/cos(x) = lim/x->0 sin(x)/cos(x) ⋅ 1/x = lim/x->0 sin(x)/x ⋅ 1/cos(X) = 1 ⋅ 1 = 1 2. lim/x->0 x/x + sin(x)/x = lim/x->0 x/x + lim/x->0 sin(x)/x = 1 + 1 = 2

Solve the following limits : 1. lim/x->0 tan(x)/x 2. lim/x->0 x+sin(x)/x

A) $80 per machine B) $90 per machine C) the more you make, the cheaper they are to make

Suppose that the cost of producing x washing machines is c(x) = 2000 + 100x - 0.1x^2 A) Find the marginal cost of producing 100 washing machines B) Find the average cost of producing the first 100 washing machines C) Compare

Horizontal Asymptote

The "K" value represents the ___.

Horizontal asymptote

The answer to a limit where x->+-infinity is the ___.

It will be a power less than the original function

The derivative of a polynomial will have what attribute?

False

The expression lim/x->a f(x) exists only if f(a) exists. True or False?

A) 3t^2 - 12t + 9 m/s B) -3 m/s and 9 m/s C) t = 3 and t = 1 D) -USE CHART- E) 6t - 12 m/s^2 F) t = 2 s

The position of a particle is represented by s = f(t) = t^3 - 6t^2 + 9t, where t is measured in seconds and s in meters. A) Find velocity. B) What is the velocity after two seconds? Four? C) When is the particle at rest? D) When is the particle moving in the positive direction? Negative? E) Find acceleration at t = 4s F) When is the particle speeding up? Slowing down?

π/2

There is a vertical asymptote at all odd multiples of ___.

1. Left hand limit -> lim/x->1- f(x) = 0 Right hand limit -> lim/x->1+ f(x) = 1 Overall limit -> DNE 2. Left hand limit -> lim/x->2- f(x) = 1 Right hand limit -> lim/x->2+ f(x) = 1 Overall limit -> lim/x->2 f(x) = 1 3. Left hand limit -> lim/x->3- f(x) = 2 Right hand limit -> lim/x->3+ f(x) = 2 Overall limit -> lim/x->3 f(x) = 2

Use *PICTURE 1* to determine the following : 1. What is the limit of the function at x = 1? Left hand limit -> Right hand limit -> Overall limit -> 2. What is the limit of the function at x = 2? Left hand limit -> Right hand limit -> Overall limit -> 3. What is the limit of the function at x = 3? Left hand limit -> Right hand limit -> Overall limit ->

y' = -80x^3 + 15x^2 + 16x - 2 Slope = -51 Equation : y = -51x + 42

Use product rule to find the equation of the tangent line to the curve of the function y = (5x^2 - 2)(x - 4x^2) at the point (1,-9).

2(x^2 - 5x +1)/(x^2 - 1)^2 NPV's : x cannot = +-1 Domain : (-infinity,-1)(-1,1)(1, infinity)

Using the quotient rule, differentiate, simplify, and state the domain. H(x) = x^2 - 2x + 4/x^2 - 1

First derivative from speed

Velocity is what form of derivative?

If the left and right limit exist, and match

What determines if there is an overall limit?