Calculus

Sketch the lines X=1, x=2, and x=3 tangent to the curve given in figure 7. Estimate the slope of each of the tangent lines you drew.

• (2 answers) The slope of the tangent line x=2 is 0. The slope of the tangent lines at x=1 is 1 and at x=3 is -1.

Use Newton's Method to find the root of x4−5x3+9x+3=0x4−5x3+9x+3=0 accurate to six decimal places in the interval [4,6].

• x ≈ 4.53

1. Write the equation of the line that represents the linear approximation to the function below at a given point a. f(x) = ln(1 + x); a = 0; f(0.9)

• y = L(x) = x

Define A(x) to be the area bounded by x and y axes, the line y=x+1, and the vertical line at x. (a) Evaluate A(2) and A(3) (b) What area would A(3) - A(1) represent?

• (a) A(2) = 4 square units A(3) = 7.5 square units (b) A(3) - A(1) = 6 square units

If f(x) and g(x) are linear functions, then f(x) + g(x) is a linear function.

• True

Use the Bisection Algorithm Method to find the root of the given function to an interval of length less than or equal to 0.1. Answer should be up to one decimal place only. f(x) = x2 - 2 on [0,3]

• 1 / 4

Use linear equation to estimate e0.06 . Choose a value of 'a' to produce a small error.

• e 0.06 = 1.06

A container in the shape of a right circular cylinder with no top has surface area 3 ft.2 What height h and base radius r will maximize the volume of the cylinder?

• r = 1 ft. h = 1 ft. V = 3.14 ft3

Given g(x) = (x+3)/(x-1). Evaluate g(5) and g(2n+1).

• g(5) = 2 g(2n+1) = 1+(2/n)

Evaluate f(3), g(-1), and h(4)

• f(3) = 1 g(-1) = -2 h(4) = 1

Let f(x) = 1-(x-1)2 evaluate (a)f(2)f(3) and (b)f(23)(a)f(2)f(3) and (b)f(23

• (a) Answer 0 (b) Answer 8/9

Build a rectangular pen with three parallel partitions using 500 feet of fencing. What dimensions will maximize the total area of the pen?

• x = 50 ft. y = 125 ft. A = 6250 ft2

Evaluate limx→0(x+1)3−1xlimx→0(x+1)3−1x

• 3

Evaluate limx→3x4−812x2−5x−3limx→3x4−812x2−5x−3

• 108 / 7

Do the following. Determine the answers by typing the missing numbers on the spaces provided. Up to two decimal places only:

• Do the following. Determine the answers by typing the missing numbers on the spaces provided. Up to two decimal places only:

(b) What was the average velocity of the car from t=20 to t=25 seconds?

• (b) average velocity = -20 feet/second

limx→13x+2=5limx→13x+2=5 What values of x guarantee that f(x) = 3x + 2 is within 0.05 unit of 5?

• If x is within 0.02 unit distance of 1, then f(x) is within 0.05 unit of 5.

The figure shows the distance of a car from a measuring position located on the edge of a straight road. (a) What was the average velocity of the car from t=0 to t= 20 sec? (b) What was the average velocity from t=10 to t=30 sec? (c) About how fast was the car traveling at t=15 sec?

• (a) V = 15 ft/sec (b) V = -5 ft/sec (c) V = 20 ft/sec

Use chain rule to calculate dydxdydx of y = tan (e3x√)(e3x)

• dydx=−sec2(e3x−−√)3e3x√23x−−√dydx=−sec2(e3x)3e3x23x

Find the slope of the line which is tangent to the circle with center C(3,1) at the point P(8,13).

• Slope of the tangent line = -5 /12

Determine whether the graph is continuous or not continuous. (GRAPH MISSING: ANSWER NOT CONFIRMED)

Find the dimensions (radius r and height h) of the cone of maximum volume which can be inscribed in a sphere of radius 2.

• r ≈ 1.89 h ≈ 2.67 V ≈ 9.93

The sum of two prime numbers is a prime.

• false

If f(x) and g(x) are linear functions then f(x)g(x) is a linear function.

• False

(Note: Answers should be in decimal form. Up to two decimal places only)

• x = 1.5 Smallest sum: S = 8.5

Evaluate limx→7x−3−−−−√limx→7x−3

• 2

Write the equation of the line that represents the linear approximation to the function below at the given point a. f(x)=e2;a=0;f(0.05)f(x)=e2;a=0;f(0.05)

• f(a) = 7.39

Use the function f defined by the graph shown to determine the following limits: (a) limx→1+f(x)limx→1+f(x) (b) limx→1−f(x)

• (a) 2 (b) -1

Determine all the critical points for the function y=6x−4cos(3x)y=6x−4cos(3x) x=???+2πn3,n=0,±1,±2,...x=???+2πn3,n=0,±1,±2,... x=???+2πn3,n=0,±1,±2,...x=???+2πn3,n=0,±1,±2,...

• 1.2217; 1.9199

An open rectangular box with square base is to be made from 48 ft.2 of material. What dimensions will result in a box with the largest possible volume?

• x = 4 ft. y = 2 ft. V = 32 ft.3

Find a linear approximation of f(x)=3xe2x−10f(x)=3xe2x−10 at x = 5

• L(x) = 15 + 33 (x - 5) = 33 x + 150

Find a value for B so that the line y = 10x – B, goes through the point (5,-5).

• B = 55

Evaluate limx→0(x+5)2−25xlimx→0(x+5)2−25x

• 10

Which of the following are integer values of x that will make the statement x>4 and x

• 5,6,7,8

Write the contrapositive of the statement: I feel good when I jog. Answer: When I don't jog, I feel

• Bad

The figure shows the distance of a car from a measuring position located on the edge of a straight road. (a) What was the average velocity of the car from t=10 to t=30 seconds?

• a) average velocity = 10 feet/second

(Note: Answers should be in decimal form only. Up to two decimal places}

• x ≈ 8.77 ft. y ≈ 16.67 ft. L ≈ 17.64 ft.

Use the linear approximation to estimate the given function value.

• f(0.05) ≈ L (0.05) = 0.05

Use chain rule to calculate dydxdydx of y=sin(4x3+3x+1)y=sin(4x3+3x+1)

• dydx=(12x2+3)cos(4x3+3x+1)dydx=(12x2+3)cos(4x3+3x+1)

Write the contrapositive of the statement: If x>3, then x2>9. Use words or phrase for your answer.

• If x2 <= 9, then x <= 3

Find the slope of the line passing through the points (3,-4) and (-6,9). Use decimal value for your final answer.

• -1.44

Let f(x) = -x 4 -x-1, evaluate f(-1) and -2f(1).

• f(-1) = -1 -2f(1) = 6

Find the line which goes through the point (2,-5) and is perpendicular to the line 3y-7x=2. (write the numerical coefficient of each term to complete the required equation)

• 3x +7y +29= 0

Every straight line on the Cartesian plane intersects the x-axis.

• True

Assume that y is a function of x. Find y1=dydxy1=dydx for cos2x+cos2y=cos(2x+2y)cos2x+cos2y=cos(2x+2y)

• y1=cosxsinx−sin(2x+2y)sin(2x+2y)−cosysinyy1=cosxsinx−sin(2x+2y)sin(2x+2y)−cosysiny

Use the function h defined by the graph shown to determine the following limits: (a) limx→2h(5−x)limx→2h(5−x) (b) limx→0h(3+x)−h(3)

• (a) 1 (b) -2

Find an equation of the line tangent to the graph of y=x2+sinπ2xy=x2+sinπ2x at x = -1

• y = -2x - 2

Use chain rule to calculate dydxdydx of y=x2sec(5x)y=x2sec(5x)

• dydx=−2xsec(5x)+5x2sec(5x)tan(5x)dydx=−2xsec(5x)+5x2sec(5x)tan(5x)

(a) limx→2h(2x−2)limx→2h(2x−2) (b) limx→2h(1+x)

• (a) 1 (b) 1

Assume that y is a function of x. Find y1=dydxy1=dydx for x−y3y+x2=x+2x−y3y+x2=x+2

• y1=1−y−3x2−4x3y2+x+2y1=1−y−3x2−4x3y2+x+2

From the figure shown, find the values of f(2), f(-1) and f(0).

• f(2) = 5 f(-1) = 2 f(0) = 1

Use the functions f and g defined by the graphs as shown to determine the following limits: (a) limx→1(f(x)xg(x))limx→1(f(x)xg(x)) (b) limx→1f(g(x))

• (a) 0 (b) 5/4

Use linear approximation to estimate the given function value

• f(2.1) = 7.6

At which values of x is the function f(x)=x2+x−6x−2f(x)=x2+x−6x−2continuous and discontinuous?

• continuous at x = -3 discontinuous at x = 2

There are 50 apple trees in an orchard. Each tree produces 800 apples. For each additional tree planted in the orchard, the output per tree drops by 10 apples. How many trees should be added to the existing orchard in order to maximize the total output of trees?

• x = 15 additional trees P =r 42250 apples

(Note: Answer should be in decimal form. Up to two decimal places only)

• x ≈ 17.32 ft. θ = 30 degrees

Evaluate limx→35x2−8x−13x2−5limx→35x2−8x−13x2−5

• 2

Compute the percent error in your approximation by the formula: |approx−exact|exact|approx−exact|exact

• Percent error: 40.22 %

Let f(x) = 2-x 2 , evaluate (a) f(x+1) and (b) f(x)+f(1)

• (a) f(x=1) = -x 2 - 2x+1 (b) f(x) + f(1) = -x 2+3

You are standing at the edge of a slow-moving river which is one mile wide and wish to return to your campground on the opposite side of the river. You can swim at 2 mph and walk at 3 mph. You must first swim across the river to any point on the opposite bank. From there walk to the campground, which is one mile from the point directly across the river from where you start your swim. What route will take the least amount of time?

• x ≈ 0.89 mi. Shortest possible time: T ≈ 0.71 hr.

Use the functions f and g defined by the graphs as shown to determine the following limits: (a) limx→1f(x)+g(x)limx→1f(x)+g(x) (b) limx→2f(x)g(x)limx→2f(x)g(x)

• (a) 2 (b) 4/3

The slope of the line through (5,15) and (x+8, x2 -2x) is

• x-5

Let f(x) = 2-x 2 , evaluate (a) f(x+1) and (b) f(x)+f(1).

• (a) f(x=1) = -x 2 - 2x+1 (b) f(x) + f(1) = -x 2+ 3

Let f(x) = (x-1)2 and define S(x) to be the slope of the line through the point (0,0) and (x,f(x)). Evaluate S(6).

• S(6) = 25/6

Use Newton's Method to find the root of 2x2+5=ex2x2+5=ex accurate to six decimal places in the interval [3,4].

• x ≈ 4.36

Use implicit differentiation to find dydxdydx, x3=x+yx−yx3=x+yx−y

• y1=3x2(x−y)2+2y2xy1=3x2(x−y)2+2y2x

Write the contrapositive of the statement: If x2 + x - 6 = 0 then x=2 or x=-3.

• Answer: If x = - 2 and x = 3 then x2 + x - 6 is not equal to 0

The process of taking the limit of a sum of little quantities is called

• Integration

(b) Evaluate A(4) - A(1)

• 9 square units

Find the slope of the line through (0,0) and (x-1, x2

• m = x+1

Which of the following equations is the line perpendicular to 2x – 3y = 9?

• 3x + 2y =10

Use the function h defined by the graph below to determine the following limits: (a) limx→2x+h(x)limx→2x+h(x) b) limx→3h(x2)

• (a) 3 (b) 3/4

Which of the following are negation of the statement: f(x) and g(x) are polynomials.

• f(x) or g(x) is a polynomial f(x) and g(x) are not polynomials

The slope of the line from point U(5,13) and the point V(x+1, x2 -3) is

• x+4

The slope of the tangent line is called the

• Derivative

The 2 divisions of Calculus are:

• Integral Differential

Which of the following figures represents the graph of a function?

• Figure 2

A __ assigns a unique output element in the range to each input element from the domain.

• function

Assume that y is a function of x. Find y1=dydxy1=dydx for y=x2y3+x3y2y=x2y3+x3y2

• y1=2xy3+3x2y21−3x2y2−2x3yy1=2xy3+3x2y21−3x2y2−2x3y

hich values of x is the function from the graph shown continuous? State the answers from the least to the highest, if there would be more than one

• x = -1

From the graph shown, find the values of f(-3), f(-1), f(0), and f(1).

• f(-3) = -1 f(-1) = 1 f(0) = 0 f(1) = 1

Which values of x will make the statement x+5=3 or x2=9 true?

• -2 or (3 and -3)

Find the slope and midpoint of the line segment from P(2,-3) to Q(2+n,-3+5n).

• Slope = 5 midpoint (0.5n+2,2.5n-3)

Let f(x)=1-(x-3)2 , evaluate: (a) f(x+3), (b) f(3-x), and (c) f(2x+1).

• (a) 1 - x 2 (b) 1 - x 2 (c) -4 x 2 + 8x-3

Evaluate limx→1x13−1x14−1limx→1x13−1x14−1

• 4 / 3

From the graph shown, find: a. f(-1) b. f(0) c. 3f(2) d. the value of x that corresponds to f(x)=0

Evaluate limx→43−x+5−−−−√x−4limx→43−x+5x−4

• 1 / 5

What is the slope of the line through (-1,-2) and (x,y) for y = x2+ 2x + 1 and x=-0.90? x=-1.05? x=h1? What happens to this last slope when h is very small? Round-off your answers to 2 decimal places whenever possible. Use the ^ symbol to express the exponent of a variable, i.e. x^2 (x squared)

• when x=-0.90: m = 2 when x=-1.05: m = 3.3 when x=h-1: m = h^2+1 /h when h approaches 0: m = 1

For the function f(x)=x(x2+1)2f(x)=x(x2+1)2 on [-2,2] Find the critical points and the absolute extreme values of f on the given interval.

• x=±13−−√x=±13 as the critical points absolute maximum value of f: 33√163316 absolute minimum value of f:33√163316

Find the local extreme values of the given function: f(x)=x4−6x2f(x)=x4−6x2

• Local minimum: (-1.73, -9) Local maximum:(1.73, -9)

Calculus was developed by Leibniz and

• Newton

Given the function f(x)=3x-4, evaluate: (a) f(x-2), (b) f(x)-f(2), (c) f(1)/f(3), and (d) f(1/3). Use fraction as final answer, if any.

• (a) 3x-10 (b) 3x-6 (c) -1/5 (d) -3

If x divides 49, then x divides 30.

• False

Given f(x) = 2x + 3 and g(x) = x2 . Evaluate . Sample text answer: 3x^2+6x7. Do not use space between the number, letter and symbol.

• 4x^2+12x+9

Write the contrapositive of the statement: If I exercise and eat right, then I will be healthy. Don't use contractions in your answer

• If I am not healthy, then I do not exercise and eat right.

For f(x) = |9-x| and g(x) = sqrt(x-1). Evaluate fog(1).

• f(g(1) = 9

Use the function h defined by the graph below to determine the following limits: (a) limx→2(xlimx→2(x . h(x−1))h(x−1)) (b) limx→0h(3+x)−h(3)h(x)limx→0h(3+x)−h(3)h(x)

• (a) 8/3 (b) -6/5

Let A = {1,2,3,4,5}, B = {0,2,4,6}, and C = {-2,-1,0,1,2,3}. Which of the values of x will satisfy each statement?

• x is in A or x is in C

If a tangent line is inclined 45 degrees, then what is the slope the tangent line?

• 1

Given g(t) = t+5t−1t+5t−1, evaluate: (a) g(5) and (b) g(2s - 5)

• (a) g(5) = 5/2 (b) g(2s-5) = s/s-3

Use chain rule to calculate dydxdydx of y=e−x2y=e−x2

• dydx=−2x−x2dydx=−2x−x2

A sheet of cardboard 3 ft. by 4 ft. will be made into a box by cutting equal-sized squares from each corner and folding up the four edges. Given that variable x shall be the length of one edge of the square cu from each corner of the sheet of cardboard, what will be the dimensions of the box with largest volume?

• x ≈ 0.57 ft, so Length = 2.86 ft Width = 1.86 ft Height = 0.57 ft V ≈ 3.03 ft

Write the negation for the statement: All quadratic equations have solutions.

• not all quadratic equations all solutions.

Use implicit differentiation to find dydxdydx (xy+1)3=x−y2+8(xy+1)3=x−y2+8

• y1=1−3y(xy+1)23x(xy+1)2+2yy1=1−3y(xy+1)23x(xy+1)2+2y

Write the negation of the statement: 8 is a prime number. 8 is

• not a prime number

Evaluate limx→0cos2x−1cosx−1limx→0cos2x−1cosx−1

• 4

Use chain rule to calculate dydxdydx of y=(5x2+11x)20y=(5x2+11x)20

• dydx=(20)(5x2+11x)19(10x+11)dydx=(20)(5x2+11x)19(10x+11)

Assume that y is a function of x. Find y1=dydxy1=dydx for y=sin(3x+4y)y=sin(3x+4y)

• y1=3cos(3x+4y)1−4cos(3x+4y)y1=3cos(3x+4y)1−4cos(3x+4y)

Find an equation describing all points P(x,y) equidistant from Q(-3,4) and R(1,-3). (use the general equation of a line

• 8x –14y +15=0

Determine all the critical points for the function. f(x)=x2ln(3x)+6f(x)=x2ln(3x)+6

• 0.20

Find the point of intersection and the angle between y = 4 - 2x and x - y = -1.

• Point of Intersection = (1 , 2 ) Angle of Intersection = -71.56 0

Find the slope of the line through (-3-1) and (x+3, y+1).

• x-5/x+6

What is the slope of the line through (3,9) and (x,y) for y=x2 and x=2.97? x=3.001? x=3+h? What happens to this last slope when h is very small (close to 0)? Round-off your answers to 2 decimal places, whenever possible.

• Slope at x=2.97 = 5.97 Slope at x=3.001 = 6.00 Slope at x=3+h = 6+h Slope when h is close to 0 = 6

Identify the absolute extrema and relative extrama for the following function. f(x)=x3f(x)=x3 on [-2,2]

• The function has an absolute maximum of 8 at x = 2 and absolute minimum of -8 at x = - 2. The function has no relative extrema.

Find the slope of the line through (-5,3) and (x+1, x-2).

• x-5/x+6

Find the equation of the line which goes through the point (3,10) and is parallel to the line 7x-y=1.

• 7x – y –11= 0

The figure shows the temperature during a day in a place. How fast is the temperature changing from 1:00 P.M. to 7:00 P.M.? Round-off your answer to 2 decimal places.

• -1.67 0F/hour

Every vertical line on the Cartesian plane intersects the x-axis.

• True

Fill in the missing the numbers to find the correct answer/s: Find two nonnegative numbers whose sum is 9 and so that the product of one number and the square of the other number is a maximum. P = xy2

• x = 3 y = 6 P = 108

Evaluate limx→103x−5−−−−−√5limx→103x−55

• 1

Find the slope and concavity of the graph pf x2y+y4=4+2xx2y+y4=4+2x at the point (-1,1)

• Slope = 4545, Concavity = downward

Find an equation of the line tangent to the graph of x2+(y−x)3=9x2+(y−x)3=9 at x=1

• y=76x+136y=76x+136

Which of the following will make the statement x2+3 > 1 true?

• x is greater than or equal to -1

2. Use linear approximation to estimate the given function value.

• f(0.9) = 0.9

Which of the following equations is the line perpendicular to 4y – 7x = 5?

• 4x + 7y – 18 = 0

Given f(x) = 2x + 3. Evaluate (f°f)(x). Sample text answer: 3x^2+6x-7. Do not use space between the number, letter and symbol.

• 4x+9

Find an equation of the line tangent to the graph of (x2+y2)3=8x2y2(x2+y2)3=8x2y2 at the point (-1,1)

• y - 1 = x + 2

Find a linear approximation to h(t)=t4−6t3+3t−7h(t)=t4−6t3+3t−7 at t=−3t=−3.

• L(t) = 227 - 267 (t + 3) = -267 t - 57

f(x) = 12 - x 2 ; a = 2 ; f(2.1)

• L(x) = -4 x + 16

Let f(x)=-1-x-2x2 , evaluate f(x+h)−f(x)hf(x+h)−f(x)h Factor out the negative sign for the final answer, if any

• -(4x+2h+1)

Let f(x) = 3x+2 and g(x) = 2x+A. Find a value for A so that f(g(x)) = g(f(x)).

• f(g(x)) = 6x+3A+2 g(f(x)) = 6x+A+4 A = 1

Use linear approximations to estimate 1–√46146. Choose a value of "a" to produce a small error

• 1–√46=f(146)≈L(146)146=f(146)≈L(146) = 12.08

Percent error ≈

• 20

The following problems could be solved by differential calculus:

• largest or smallest volume of a solid rate or speed

Determine all the critical points for the function. f(x)=xex2

• does not have any critical points

What is the slope of the line through (2,4) and (x,y) for y = x2+ x - 2 and x=1.99? x=2.004? x=2+h. What happens to this last slope when h is very small?

• when x=1.99: m = 4.99 when x=2.004: m = 5.00 when x=2+h:m = 5+h when h approaches 0: m = 5

For f(x) = 3x-2 and g(x) = x2+1, find the composite function defined by f o g(x) and g o f(x).

• f o g(x) = 3x^2+1 g o f(x) = 9x^2-12x+5

Find the equation of a circle with radius=6 and center C(2,-5). (write the required exponent after the ^ symbol; write the numerical coefficient of each term to complete the required equation)

• X ^2+ y ^2–4x +10y –7= 0

Find the equation of the line passing through (-2,3) and perpendicular to the line 4x=9-2y. Use the general equation of the line for your final answer.

• X –2y +8= 0

What values of x will make the statement x+5=3 or x2=9.

• x = -2 or (x = 3 and x = -3 )

Use the graph below to determine the right-hand limit of the function f(x) at: (a) x=-2 (b) x=10

• (a) undefined (b) 0

Find the slope of the line through (0,0) and (x-1, x2 -1).

• m = x+1

Find the length and midpoint of the interval from x=9 to x=-2. (use decimal values for fractional answer)

• Length =11and midpoint =3.5

Find the point of intersection and the angle between x - y = 32 and 3x - 8y = 6.

• Point of Intersection = (50 , 18 ) Angle of Intersection = -24.44 0 (round-off to 2 decimal places)

Use implicit differentiation to finddydxdydx exy=2yexy=2y

• y1=yexy2−xexyy1=yexy2−xexy

Refer to the figure. Which of the following represents the graph drawn in red? Select one:

• g(x)-1

Find the point of intersection and the angle between 2x - 3y = 3 and 4x - 2y = 10.

• Point of Intersection = (3 , 1 ) Angle of Intersection = 29.740

If f(x) and g(x) are linear functions, the f(x) + g(x) is a linear function

• True

The slope of a horizontal line is

Use Newton's Method to determine x2x2 for f(x)=xcos(x)−x2f(x)=xcos(x)−x2 if x0=1x0=1

• x2 = 0.74

If a and b are real numbers, then (a+b)2 = a2+b2 .

• False

Use chain rule to calculate dydxdydx of y=cos4(7x3)y=cos4(7x3)

• dydx=−84x2cos3(7x3)sin(7x3)dydx=−84x2cos3(7x3)sin(7x3)

Assume that y is a function of x. Find y1=dydxy1=dydx for (x−y)2=x+y−1(x−y)2=x+y−1

• y1=2y−2x+12y−2x−1y1=2y−2x+12y−2x−1

Given f(x) = x3 - 4x2 +2, f(2) when evaluated is

• -6

From the figure shown, A(x) is defined to be the area bounded by the x and y axes, the horizontal line y=3 and the vertical line at x. For example A(4)=12 is the area of the 4 by 3 rectangle (a) Evaluate A(2.5)

• 7.5 square units

The graph shows the population growth of bacteria on a petri plate. If at t=10 days, the population grows to 4600 bacteria, find the rate of population growth from t=9 to t= 10 days?

• rate of growth = 400

A function f is given by f(7-11x) = 3x3 - 10x. Evaluate f(-4).

• f(-4) = -7

(Large lim_{x rightarrow 3} (2x + 1) = 7 ) What values of x guarantee that f(x) = 2x + 1 is within 0.04 units of 7? If x is within ­­­­­­_____ units of 3, then f(x) is within 0.04 units of 7.

• 0.02

For all positive real numbers a and b, if a > b, then a2 > b2

• TRUE

If a and b are real numbers then (a + b)2 = a2 + b2

• False

Consider a rectangle of perimeter 12 inches. Form a cylinder by revolving this rectangle about one of its edges. What dimensions of the rectangle will result in a cylinder of maximum volume?

• r = 4 ft h = 2 ft V ≈ 100.53 ft3

Use the function h defined by the graph shown to determine the following limits: (a) limx→2h(5−x)limx→2h(5−x) (b) limx→0h(3+x)−h(3)limx→0h(3+x)−h(3

• (a) 3 (b) 3/4

Given a function f, an interval [a,b] and a value V. Find a value c in the interval so that f(c)=V. Apply the Intermediate Value Theorem. (a)f(x)=x2f(x)=x2 on [0,3], V = 2 (b)f(x)=sinx on [0,π2],V=12f(x)=sinx on [0,π2],V=12

• (a) c = -1.41 ; c = 1.41 (b) c = 0.52

Which of the following is the contrapositive for the statement: If your car is properly tuned, it will get at least 24 miles per gallon.

• If your car will not get at least 24 miles per gallon, then it is not properly tuned.

Locate the critical points of the following functions. Then use the second derivative test to determine whether they correspond to local minima or local maxima or whether the test is inconclusive.

• Critical points: (2, -1/4) and (10, -1/20) Local minimum: x = -2 Local maximum: x = 10