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Mathematics branch that studies change and motion, providing tools and techniques for analyzing and solving problems involving rates of change and accumulation.
Let f(x) = 2-x 2 , evaluate (a) f(x+1) and (b) f(x)+f(1).
Find the slope of the line which is tangent to the circle with center C(3,1) at the point P(8,13).
Find the slope of the line through (-5,3) and (x+1, x-2).
Which of the following are integer values of x that will make the statement x>4 and x
Which of the following figures represents the graph of a function?
(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.
Find the slope and midpoint of the line segment from P(2,-3) to Q(2+n,-3+5n).
Write the contrapositive of the statement: I feel good when I jog. Answer: When I don't jog, I feel
Let f(x)=1-(x-3)2 , evaluate: (a) f(x+3), (b) f(3-x), and (c) f(2x+1).
Which of the following equations is the line perpendicular to 4y – 7x = 5?
(b) Evaluate A(4) - A(1)
Find the slope of the line through (-3-1) and (x+3, y+1).
For all positive real numbers a and b, if a > b, then a2 > b2
The following problems could be solved by differential calculus:
Use chain rule to calculate dydxdydx of y=(5x2+11x)20y=(5x2+11x)20
Assume that y is a function of x. Find y1=dydxy1=dydx for y=x2y3+x3y2y=x2y3+x3y2
Find an equation of the line tangent to the graph of y=x2+sinπ2xy=x2+sinπ2x at 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)
Find the point of intersection and the angle between x - y = 32 and 3x - 8y = 6.
Evaluate limx→35x2−8x−13x2−5limx→35x2−8x−13x2−5
The slope of a horizontal line is
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)
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?
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.
Find a linear approximation of f(x)=3xe2x−10f(x)=3xe2x−10 at x = 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?
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]
Write the negation for the statement: All quadratic equations have solutions.
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
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)
If a and b are real numbers then (a + b)2 = a2 + b2
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.
A function f is given by f(7-11x) = 3x3 - 10x. Evaluate f(-4).
Let f(x) = 2-x 2 , evaluate (a) f(x+1) and (b) f(x)+f(1)
Determine all the critical points for the function. f(x)=x2ln(3x)+6f(x)=x2ln(3x)+6
Use chain rule to calculate dydxdydx of y=x2sec(5x)y=x2sec(5x)
Use the linear approximation to estimate the given function value.
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))
Use the graph below to determine the right-hand limit of the function f(x) at: (a) x=-2 (b) x=10
Find the local extreme values of the given function: f(x)=x4−6x2f(x)=x4−6x2
Write the contrapositive of the statement: If x>3, then x2>9. Use words or phrase for your answer.
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)
Write the contrapositive of the statement: If I exercise and eat right, then I will be healthy. Don't use contractions in your answer
Use implicit differentiation to finddydxdydx exy=2yexy=2y
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?
Use implicit differentiation to find dydxdydx (xy+1)3=x−y2+8(xy+1)3=x−y2+8
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
Find the slope of the line through (0,0) and (x-1, x2 -1).
Which values of x will make the statement x+5=3 or x2=9 true?
Determine all the critical points for the function. f(x)=xex2
(Note: Answers should be in decimal form. Up to two decimal places only)
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?
For f(x) = |9-x| and g(x) = sqrt(x-1). Evaluate fog(1).
2. Use linear approximation to estimate the given function value.
Percent error ≈
Let f(x) = 3x+2 and g(x) = 2x+A. Find a value for A so that f(g(x)) = g(f(x)).
Evaluate limx→103x−5−−−−−√5limx→103x−55
If f(x) and g(x) are linear functions, the f(x) + g(x) is a linear function
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?
(b) What was the average velocity of the car from t=20 to t=25 seconds?
Find the equation of the line which goes through the point (3,10) and is parallel to the line 7x-y=1.
A __ assigns a unique output element in the range to each input element from the domain.
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
Build a rectangular pen with three parallel partitions using 500 feet of fencing. What dimensions will maximize the total area of the pen?
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).
Calculus was developed by Leibniz and
Find the length and midpoint of the interval from x=9 to x=-2. (use decimal values for fractional answer)
Use chain rule to calculate dydxdydx of y=cos4(7x3)y=cos4(7x3)
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?
Find an equation of the line tangent to the graph of (x2+y2)3=8x2y2(x2+y2)3=8x2y2 at the point (-1,1)
Evaluate limx→0(x+5)2−25xlimx→0(x+5)2−25x
Identify the absolute extrema and relative extrama for the following function. f(x)=x3f(x)=x3 on [-2,2]
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?
Given g(x) = (x+3)/(x-1). Evaluate g(5) and g(2n+1).
Find the slope of the line passing through the points (3,-4) and (-6,9). Use decimal value for your final answer.
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)
Evaluate limx→0(x+1)3−1xlimx→0(x+1)3−1x
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)
Given g(t) = t+5t−1t+5t−1, evaluate: (a) g(5) and (b) g(2s - 5)
Every straight line on the Cartesian plane intersects the x-axis.
Write the contrapositive of the statement: If x2 + x - 6 = 0 then x=2 or x=-3.
Find the dimensions (radius r and height h) of the cone of maximum volume which can be inscribed in a sphere of radius 2.
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
At which values of x is the function f(x)=x2+x−6x−2f(x)=x2+x−6x−2continuous and discontinuous?
From the figure shown, find the values of f(2), f(-1) and f(0).
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).
Which of the following are negation of the statement: f(x) and g(x) are polynomials.
Find the point of intersection and the angle between 2x - 3y = 3 and 4x - 2y = 10.
Use Newton's Method to find the root of 2x2+5=ex2x2+5=ex accurate to six decimal places in the interval [3,4].
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].
Refer to the figure. Which of the following represents the graph drawn in red? Select one:
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)
The 2 divisions of Calculus are:
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?
(a) limx→2h(2x−2)limx→2h(2x−2) (b) limx→2h(1+x)
Evaluate limx→1x13−1x14−1limx→1x13−1x14−1
Evaluate f(3), g(-1), and h(4)
Assume that y is a function of x. Find y1=dydxy1=dydx for y=sin(3x+4y)y=sin(3x+4y)
If a and b are real numbers, then (a+b)2 = a2+b2 .
Write the negation of the statement: 8 is a prime number. 8 is
Evaluate limx→3x4−812x2−5x−3limx→3x4−812x2−5x−3
Assume that y is a function of x. Find y1=dydxy1=dydx for cos2x+cos2y=cos(2x+2y)cos2x+cos2y=cos(2x+2y)
Use chain rule to calculate dydxdydx of y = tan (e3x√)(e3x)
Find an equation of the line tangent to the graph of x2+(y−x)3=9x2+(y−x)3=9 at x=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
If x divides 49, then x divides 30.
Use linear approximations to estimate 1–√46146. Choose a value of "a" to produce a small error
If f(x) and g(x) are linear functions then f(x)g(x) is a linear function.
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.
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.
Evaluate limx→0cos2x−1cosx−1limx→0cos2x−1cosx−1
Assume that y is a function of x. Find y1=dydxy1=dydx for x−y3y+x2=x+2x−y3y+x2=x+2
(Note: Answers should be in decimal form only. Up to two decimal places}
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.
If a tangent line is inclined 45 degrees, then what is the slope the tangent line?
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)
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.
Which of the following equations is the line perpendicular to 2x – 3y = 9?
If f(x) and g(x) are linear functions, then f(x) + g(x) is a linear function.
Use linear equation to estimate e0.06 . Choose a value of 'a' to produce a small error.
limx→13x+2=5limx→13x+2=5 What values of x guarantee that f(x) = 3x + 2 is within 0.05 unit of 5?
(Note: Answer should be in decimal form. Up to two decimal places only)
Use Newton's Method to determine x2x2 for f(x)=xcos(x)−x2f(x)=xcos(x)−x2 if x0=1x0=1
Let f(x) = -x 4 -x-1, evaluate f(-1) and -2f(1).
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?
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
The sum of two prime numbers is a prime.
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
Compute the percent error in your approximation by the formula: |approx−exact|exact|approx−exact|exact
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?
Use implicit differentiation to find dydxdydx, x3=x+yx−yx3=x+yx−y
Use chain rule to calculate dydxdydx of y=e−x2y=e−x2
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.
Find the point of intersection and the angle between y = 4 - 2x and x - y = -1.
The slope of the line from point U(5,13) and the point V(x+1, x2 -3) is
What values of x will make the statement x+5=3 or x2=9.
Which of the following will make the statement x2+3 > 1 true?
Find the slope of the line through (0,0) and (x-1, x2
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?
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.
Find a linear approximation to h(t)=t4−6t3+3t−7h(t)=t4−6t3+3t−7 at t=−3t=−3.
Evaluate limx→7x−3−−−−√limx→7x−3
The slope of the line through (5,15) and (x+8, x2 -2x) is
Find a value for B so that the line y = 10x – B, goes through the point (5,-5).
Every vertical line on the Cartesian plane intersects the x-axis.
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)
The slope of the tangent line is called the
Use linear approximation to estimate the given function value
Do the following. Determine the answers by typing the missing numbers on the spaces provided. Up to two decimal places only:
Find the slope and concavity of the graph pf x2y+y4=4+2xx2y+y4=4+2x at the point (-1,1)
The process of taking the limit of a sum of little quantities is called
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?
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,...
From the graph shown, find the values of f(-3), f(-1), f(0), and f(1).
f(x) = 12 - x 2 ; a = 2 ; f(2.1)
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.
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)
Use chain rule to calculate dydxdydx of y=sin(4x3+3x+1)y=sin(4x3+3x+1)
Evaluate limx→43−x+5−−−−√x−4limx→43−x+5x−4
Determine whether the graph is continuous or not continuous. (GRAPH MISSING: ANSWER NOT CONFIRMED)
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.
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)
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
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
Given f(x) = x3 - 4x2 +2, f(2) when evaluated is
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