Calculus 1 MATH-6100 exam answers

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 Correct

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 Correct
f(0) = 1 Correct
3f(2) = -14
x = 0

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 Correct

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 Correct

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 Correct

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

-6 Correct

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 Correct

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

g(5) = 2 g(2n+1) = 1+(2/n) Correct

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 Correct

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 Correct

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. Correct

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

False Correct

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

1 Correct

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

False Correct

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

True Correct

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

True Correct

If x divides 49, then x divides 30.

False Correct

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 Correct

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

f(-1) = -1 -2f(1) = 6 Correct

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 Correct

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 Correct

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 Correct

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 Correct

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 Correct

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) Correct

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 Correct

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. Correct

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 Correct

Percent error ≈

20 Correct

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

g(x)-1 Correct

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. Correct

The 2 divisions of Calculus are:

Integral Differential Correct

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 Correct

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 Correct

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 Correct

The following problems could be solved by differential calculus:

largest or smallest volume of a solid rate or speed Correct

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 Correct

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

Integration Correct

The slope of a horizontal line is

0 Correct

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

x+4 Correct

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

x-5 Correct

The slope of the tangent line is called the

Derivative Correct

The sum of two prime numbers is a prime.

false Correct

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 Correct

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

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

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) Correct

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

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

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

dydx=−2x−x2dydx=−2x−x2 Correct

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) Correct

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) Correct

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 Correct

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

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

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

y1=yexy2−xexyy1=yexy2−xexy Correct

Use linear approximation to estimate the given function value

f(2.1) = 7.6 Correct

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 Correct

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

e 0.06 = 1.06 Correct

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

x2 = 0.74 Correct

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 Correct

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 Correct

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 Correct

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 Correct

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 Correct

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 Correct

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 Correct

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 Correct

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 Correct

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 Correct

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 Correct

Use the linear approximation to estimate the given function value.

f(0.05) ≈ L (0.05) = 0.05 Correct

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 Correct

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 Correct

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 Correct

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

x = -2 or (x = 3 and x = -3 ) Correct

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

5,6,7,8 Correct

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 Correct

Calculus 1

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