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Calculus

Mathematics branch that studies change and motion, providing tools and techniques for analyzing and solving problems involving rates of change and accumulation.

differentiation

integration

limits

derivatives

antiderivatives

functions

tangent

rate

slope

area

curve

optimization

inflection

convergence

divergence

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

  • x-5/x+6

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

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

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

  • 3

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.

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

(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.

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

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

  • x is greater than or equal to -1

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

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

  • 108 / 7

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

  • x = 1.5 Smallest sum: S = 8.5

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

  • x2 = 0.74

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

  • -1.44

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

  • 4x + 7y – 18 = 0

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

  • -6

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 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

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

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

  • x-5

Calculus was developed by Leibniz and

  • Newton

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

  • 5,6,7,8

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

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

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

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

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

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)

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.

Evaluate limx→7x−3−−−−√limx→7x−3

  • 2

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

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.

If x divides 49, then x divides 30.

  • False

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

  • 3x + 2y =10

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

  • Not Continuous

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

  • 1

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

  • False

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

  • L(x) = 15 + 33 (x - 5) = 33 x + 150

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

  • function

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

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)

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

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

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

  • f(3) = 1 g(-1) = -2 h(4) = 1

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

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

The slope of a horizontal line is

  • 0

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

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

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

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)

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

  • x = -2 or (x = 3 and x = -3 )

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

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

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

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

(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

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

  • Integration

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 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

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.

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

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

  • 4

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

  • f(-4) = -7

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

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:

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

  • True

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)

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

  • 1

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

  • y1=yexy2−xexyy1=yexy2−xexy

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

  • True

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

  • y = -2x - 2

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

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

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

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

  • not all quadratic equations all solutions.

Use the linear approximation to estimate the given function value.

  • f(0.05) ≈ L (0.05) = 0.05

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

The 2 divisions of Calculus are:

  • Integral Differential

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

The sum of two prime numbers is a prime.

  • false

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

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

  • 0.20

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

  • -2 or (3 and -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

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

  • 1 / 5

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

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

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

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

  • 10

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

  • f(2) = 5 f(-1) = 2 f(0) = 1

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

  • does not have any critical points

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

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

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

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

  • (a) 1 (b) 1

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

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

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

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

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

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

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

  • True

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

  • g(5) = 2 g(2n+1) = 1+(2/n)

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

  • B = 55

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

  • m = x+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

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

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

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)

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

  • Length =11and midpoint =3.5

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

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.

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

  • 9 square units

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

  • y1=3x2(x−y)2+2y2xy1=3x2(x−y)2+2y2x

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)

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

  • e 0.06 = 1.06

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

  • L(x) = -4 x + 16

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

  • False

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

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

  • dydx=−2x−x2dydx=−2x−x2

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

  • Percent error: 40.22 %

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

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

  • g(x)-1

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

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

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

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

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

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

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

  • x ≈ 17.32 ft. θ = 30 degrees

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

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

  • x-5/x+6

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)

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

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

  • m = x+1

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

  • True

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

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

Percent error ≈

  • 20

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

  • f(-1) = -1 -2f(1) = 6

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

  • False

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

  • f(-1) = 2
  • f(-1) = 2
  • f(0) = 1

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

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

  • 4 / 3

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

  • TRUE

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

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.

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

  • not a prime number

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

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

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

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

  • f(g(1) = 9

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

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

  • 2

Use linear approximation to estimate the given function value

  • f(2.1) = 7.6

The following problems could be solved by differential calculus:

  • largest or smallest volume of a solid rate or speed

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)

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

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

  • Figure 2

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

  • f(0.9) = 0.9

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

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

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
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