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