Calc 2 Exam 1

hyperbolic sine of Pi/4 equals to ... (pick the correct number)

( exp(Pi/4)-exp(-Pi/4) )/2

The equation of the line tangent to a graph y=f(x) of a function f through a point (a, f(a) ) , where a is in the domain of both f and f' , is given by the formula (A) y = f(a) + m*(x-a) where the constant m = f'(a) = derivative of f wrt x evaluated at x=a (B) y - f(a) = m*(x-a) where the constant m = f'(a) = derivative of f wrt x evaluated at x=a (C) For x not exactly equal to a, the equation is: m = (y - f(a)) / (x - a) = f'(a) = derivative of f wrt x evaluated at x=a, while for x=a the point on the straight line is (a, f(a)). (D) y = mx+b where b = f(a) - a*m, the constant m = f'(a) = derivative of f wrt x evaluated at x=a (E) m = f'(a) = derivative of f wrt x evaluated at x=a (F) y - f(a) = f'(x)*(x-a) where f'(x) = df/dx = derivative of f wrt x (G) y = f(a) + f'(x)*(x-a) where f'(x) = df/dx = derivative of f wrt x

(A), (B), (C), (D) are all correct, representing different algebraic forms of the same equation

cos^2(x) - sin^2(x) = ? Pick the expression from the selection to use in the right hand side which makes the statement above an identity (A) 1 (B) sin(2x) (C) cos(2x) (D) None of the above

(C) cos(2x)

(lim (cos(x) - 1)/x^2, x-> 0 ) =?

-1/2

(lim (cos(x) -1)/x , x-> 0 ) = ?

0

Find the area under the curve y = 1/(4 + 9*x2) on [0, 1/3]

0.077275

Find the area under the curve y = 1/(4 - 9*x2) on [0, 1/3]

0.09155

Find the area under the curve y = 1/sqrt(4 +9*x2) on [0, 1/3]

0.16040

Find the area under the curve y = 1/sqrt(4 - 9*x2) on [0, 1/3]

0.17453

Estimate sin(1.5) = cos(Pi/2 - 1.5) using quadratic (in x ) approximation for cos(x) centered at zero, that you have derived in the Questions 1 and 2. Note , (Pi/2 - 1.5) is in proximity of zero. Assume Pi accurate to 9 digitsL 3.14159265 Round your answer to 6 digits

0.997494

Find the area under the Cartesian graph of a function f(x) = sin(2*x) on the interval [0, Pi/2]

1

Let the limit be: limit (arctan(x))^x, x->0 Evaluate the limit using technology.

1

limx->0 sin(x)/x = ?

1

Use the tangent line to y=exp(x) through (0, exp(0)) = (0,1) to estimate: (the tenth root of e) = e^(1/10)=exp(0.1)

1.1

d( arcsin(x) )/dx , also denoted (arcsin(x))', equals (pick the correct response)

1/sqrt(1-x^2)

Let sinh-1(1) = ln (Q). Find Q (either exact or 2 points after the decimal) Steps: Express sinh-1(x) using a known formula as a logarithm

2.41

2^3^^2 stands for (pick the correct answer)

2^9 = 512

nverse hyperbolic sine of 3 is natural logarithm of which number?

3+sqrt(10

Which number equals ln(5)? ln(10) - ln(2) ln(10) / ln(2) log 2 (5) log 10 (5)

A) ln(10) - ln(2)

Complete the coefficient in the polynomial approximations for arctan(x) below. Use your results in (1a, 1b, 1c) to find the coefficients. arctan(x) is approximated as A*x +B*x^2+C*x^3 near a=0 What are the values for the constant coefficients A, B, C?

A=1, B=0, C=-1/3

If we approximate (arctan(0.12) ) radians with the number (0.12 - 0.12^3/3) would it be an over-approximation or an under-approximation?

An underestimate: the number (0.12 - 0.12^3/3) is smaller than the actual value of (arctan(0.12) )

A)The graph features log-log (both x-log and y-log) scales. The base a=10 is used. B)The graph features a semi-logarithmic (y-log) scale. The base a=10 is used. C)The graph does not feature a logarithmic scale and thus the question about the base is not relevant. D)The graph features a semi-logarithmic (y-log) scale. The base a=100 is used. E)The graph features a semi-logarithmic (x-log) scale. The base a=50 is used.

B) The graph features a semi-logarithmic (y-log) scale. The base a=10 is used.

Which number equals log 2 (5)? ln(10) - ln(2) ln(5)/ln(2) ln(10)/ln(2) None of the displayed answers

B) ln(5)/ln(2)

Which number equals log 2 (10)? ln(10) - ln(2) ln(5)/ln(2) ln(10)/ln(2) None of the displayed answers

C) ln(10)/ln(2)

Two cars have velocities that are modeled as: Car A: va(t) = 4* sin(3t) Car B: vb(t) = 4* sinh(t) Which car moves faster at the moment t=0.2?

Car B

Find the explicit function and graph in the Cartesian plane the inverse of the following function

D (the formula for the inverse function is x^2/2-5/2 and the domain of the inverse function is x>=0.)

Your team of students evaluates the integral below, and you are the team's captain. You must judge which opinion is the best and submit your report. Which opinion would you choose? integral sin(2*x) dx = ? Amy says: the answer is sin^2(x) + const Bob says: the answer is (-1/2)*cos(2x) + const Caleb says: Either Amy or Bob or both of them are wrong, because sin^2(x) is not the same, as (-1/2)*cos(2x) Diana says: the correct answer is ( - cos^2(x) + const) Elias says: All of you are wrong because integration must result in a specific real number Fiona says:I agree with Amy, Bob, and Diana: all three of them are correct

Fiona

Let the function be f(x) = arcsin(x) - sinh-1(x). Does the function increase or decrease at the instant x = 0.5? Steps: Find f'(x) and f'(0.5)

Increases

Joe estimates sin(0.6 rad) as 0.6 Jill estimates sin(0.06 rad) as 0.06 Linda estimates sin (34o) as 34 Rank the quality of their estimates, listing the estimate with the lowest percentage error first, and the estimate with the highest percentage error last

Jill, Joe, Linda

Tom wants to see, how Newton's method for roots responds to a "bad" initial condition. He sets the initial condition to be very far from his target sqrt(5): r(0)=500. Tom implements the IVP RR: r(n+1)=(r(n)+5/r(n))/2, r(0)=500 and runs it till, for some n=N, r(N) approximates sqrt(5) accurate to 5 digits. At minimum, how many steps does it take: N=?

N=11

Evalute the area between curves y=sin(2x) and y=sin(x) on [0,Pi/2] integral abs(sin(2*x) - sin(x))) dx, x=0...Pi/2 = ?

None of the displayed answers is correct

Dan approximates sine of (0.6 rad) as 0.6, it is an overestimate or underestimate?

Overestimate because the tangent line through the origin "overshoots" over the true value

A)The right graph features a semi-logarithmic (x-log) scale with the data scope compression consistent with the usage of the base a=2 logarithms. B)The left graph features log-log (both x-log and y-log) scales with the data scope compression consistent with the usage of the base a=2 logarithms. C)Both graphs feature the x-log scale with the base of 10 used in the background formula. D)There are no logarithmic scales on any of these graphs.

The right graph features a semi-logarithmic (x-log) scale with the data scope compression consistent with the usage of the base a=2 logarithms.

The two most fundamental problems which Calculus solves (and pre-Calculus does not) are commonly referred to as...

The tangent line problem and the area problem

Use L'Hospital's Rule to evaluate: (1a) limit arctan(x)/x, x->0 (1b) limit (arctan(x)-x)/x^2, x->0 (1c) limit (arctan(x)-x)/x^3, x-> 0 (1d) limit (arcsin(x)-x)/x^3, x-> 0 Please check your answers with technology before submitting them. The limits in the problems are stated in a form that may be used as a direct input in wolframalpha.com

a.) 1 b.) 0 c.) -1/3 d.) 1/3! = 1/6

sin-1x symbol means (pick the correct symbol)

arcsin(x)

cos2x - sin2x identically equals to (pick the correct response )

cos(2x)

Mike collects functions which share the tangent line through the origin given by "y=x" (the bisector of the 1st and the 3rd quadrants). He makes a list, but his list contains one WRONG function, which does not feature this tangent line at the origin. Help Mike: which function does NOT belong on his study card? sin(x), cos(x), tan(x), arcsin(x), arctan(x)

cos(x)

If x>0 is near a=0, sin(x) is overestimated with x, and exp(x) is underestimated with (1+x) Which estimate has a bigger absolute error for x=0.6?

exp(0.6) estimated as 1.6 has a bigger absolute error than sin(0.6 rad) estimated with 0.6 because exponential curves away from a straight line more than sine function's graph does in the neighborhood of a=0

Roots and Natural logarithm from the inverse function perspective. Applications of natural logarithm question 2 image

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Roots and Natural logarithm from the inverse function perspective. Applications of natural logarithm question 3 image

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Roots and Natural logarithm from the inverse function perspective. Applications of natural logarithm question 4 image

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