Calculus 1 MATH-6100 exam answers

(\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 Correct

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

(a) 1 (b) 1 Correct

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

9 square units Correct

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

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

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

x ≈ 17.32 ft. θ = 30 degrees Correct

(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. Correct

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

x = 1.5 Smallest sum: S = 8.5 Correct

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 Correct

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

f(0.9) = 0.9 Correct

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

function Correct

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 Correct

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

f(-4) = -7 Correct

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 Correct

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 Correct

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 Correct

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 Correct

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 Correct

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) Correct

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 Correct

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 Correct

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 Correct

Calculus was developed by Leibniz and

Newton Correct

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

Percent error: 40.22 % Correct

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 Correct

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 Correct

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 Correct

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

0.20 Correct

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

does not have any critical points Correct

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

Not Continuous

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: Correct

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

f(3) = 1 g(-1) = -2 h(4) = 1 Correct

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

3 Correct

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

10 Correct

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

4 Correct

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

1 Correct

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

4 / 3 Correct

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

2 Correct

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

108 / 7 Correct

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

1 / 5 Correct

Evaluate limx→7x−3−−−−√limx→7x−3

2 Correct

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

True Correct

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

True Correct

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

L(x) = -4 x + 16 Correct

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 Correct

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

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

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 Correct

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

B = 55 Correct

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 Correct

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 Correct

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 Correct

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

y = -2x - 2 Correct

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 Correct

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 Correct

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 Correct

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 Correct

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 Correct

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 Correct

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) Correct

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 Correct

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) Correct

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 Correct

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 Correct

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) Correct

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

-1.44 Correct

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

x-5/x+6 Correct

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

x-5/x+6 Correct

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

m = x+1 Correct

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

m = x+1 Correct

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 Correct

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

TRUE Correct

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

f(g(1) = 9 Correct

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 Correct

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 Correct

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 Correct

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

f(2) = 5 f(-1) = 2 f(0) = 1 Correct

Calculus 1

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