CALC FINAL 1552 TRUE/FALSE

If F and G are both anti-derivatives of the continuous function f, then F(x) = G(x).

False

Let F(x) = integral from a(x) to b of f(t) dt. Then F’(x) = -f(a(x)) * a’(x)

True

Suppose f is a continuous function on [a,b] and the integral from a to b of f(x) dx = 0.  Then f(x) = 0 for all x E [a,b].

False

Suppose that f and g are both continuous on [a,b], and that f(x) > g(x). Then the integral from a to b of |f(x)| dx > the integral from a to b |g(x)| dx.

False

If f is a continuous, increasing function, then the right-hand Riemann sum method always overestimates the definite integral.

True

Let f be a continuous function and av(f) be the average of f. Then av(f) * (b-a) = the integral from a to b of f(x) dx.

True

If the integral from 0 to 1 of f(x) dx = 9 and f(x) >= 0, then the integral from 0 to 1 of square root of f(x) dx = 3.

False

integral of tan(x) dx = sec²(x) + C

False

inetegral of e^ax dx = 1/(aln(e)) * e^ax + C

True

We always have Uf >= Lf

True

We always have |Uf| >= |Lf|

False

1+2+3+…+100 = 5050

True

If the integral from 4 to 5 of f(x) dx = 2 then the integral from -5 to -4 of -f(x) dx = -2

False

The Fundamental Theorem of Calculus lets us compute indefinite integrals.

False

Integration by substitution is often called “𝑢-sub” because you must use the letter 𝑢 as the substitute variable.

False

The method of integration by substitution is the reverse of the chain rule for derivatives

True

If 𝑓(𝑦) is always greater than 𝑔(𝑦) then the area bounded by 𝑓, 𝑔, 𝑦 = 𝑎 and 𝑦 = 𝑏 can be found by evaluating the integral from a to b of 𝑓(𝑦) − 𝑔(𝑦) 𝑑𝑦

True

If we do by-parts and get back what we started with then we have to try a different method.

False

In by-parts we take the derivative of 𝑢 to get 𝑑𝑢 and take the integral of 𝑑𝑣 to get 𝑣.

True

sin²(x) = 1 -cos²(x)

True

sin²(x) = ½ - ½ sin(2x)

False

To find sin 𝜃 in a triangle, divide the length of the opposite side by the length of the adjacent side

False

We can use the method of trig sub to solve the integral of square root of 1 − 9𝑡² 𝑑𝑡.

True

In a fraction of polynomials, if the degree of the numerator equals the degree of the denominator, we must do long division

True

If Alice does partial fractions and gets 𝐴 = 2, then when Bob does partial fractions, he will also get 𝐴 = 2

False

L’Hopital’s rule tells us that for any functions 𝑓 and 𝑔, if lim𝑥→𝑎 𝑓′(𝑥)/𝑔′(𝑥) = 𝐿 then lim𝑥→𝑎 𝑓(𝑥)/𝑔(𝑥) = 𝐿

False

If the integral from 0 to inf of f(x) dx converges then both the integral from 0 to 1 of f(x) dx and the integral from 1 to inf of f(x) dx must converge

True

A sequence cannot have more than one g.l.b.

True

The 𝑛th term test cannot prove that a series converges.

True

When determining if an infinite series converges, we recommend to always try the integral test first.

False

We can always use the limit comparison test instead of the basic comparison test

False

A ratio test and root test that result in the same limit will give the same conclusion

True

If a series has factorials then it is a good candidate for the root test.

False

If the alternating series ∑ from n=1 to inf of an diverges but ∑ from n=1 to inf |an| converges, then ∑ from n=1 to inf of an converges conditionally.

False

It is possible for a series to be alternating even if it does not include the term (−1)^n.

True

If p > 1, the alternating p-series of ∑ from n=1 to inf of (-1)^n-1 / n^p coneverges conditionally

False

The alternating harmonic series ∑ from n=1 to inf of (-1)^n+1 1/n is conditionally convergent

True

The function f(x) will always have a smaller radius of convergence than f(2x)

False

If the function f(x) has the MacLaurin series f(x) = ∑ from k=0 to inf akx^k, then the Taylor series of f(x) centered at c is given by f(x) = ∑ from k=0 to inf of ak(x-c)^k

False

If the radius of convergence of a power series is 0, then the power series diverges for every real number

False

If a power series ∑ from n=0 to inf of anx^n has a radius of convergence of 1, then the interval of convergence is (-1,1)

False

Every Taylor series is a power series

True

The fifth degree Taylor Polynomial for cos(x) about x = 0 is 1 - x²/2 + x^4/4!.

True

∑ from k=0 to inf of (-2)^k / k! = -e²

False

The Polynomial P(x) = -5/2(x-1/2)+(x-1/2)³ could be the 5th-order Taylor polynomial centered at x=1/2 of a function f(x).

True

If the power series ∑ n of cnx^n converges at x=3, then it also converges at x=-3

False

A series representation for the definite integral from 0 to x of e^t² dt is ∑ from k=0 to inf of 1/(2k+1)*k! * x^2k+1

True

If a certain value of 𝑥 causes a power series to converge conditionally, we include it in the interval of convergence

True

It is possible to get an interval of convergence that is empty

False

The only way to make a Taylor polynomial approximation more accurate is to use a higher degree.

False

To find the error of an alternating series, we should use the Taylor Remainder Theorem.

False

All Maclaurin series are Taylor series

True

All power series are Taylor series.

False

The shell method is based on the formula for the surface area of a cylinder: 𝑆𝐴 = 2𝜋𝑟ℎ.

True

The line 𝑥 = 2 is parallel to the 𝑥-axis.

False

If F and G are two functions with the same derivative, then F = G.

False

The total area under the curve y = x² bounded by the lines x = -2 and x = 1 is given by the integral from -2 to 1 of x² dx

True

The antiderivative of an even function is even.

False

We can evaluate the integral sec² (x) tan³ (x) dx by substitution, setting u = sec (x).

False

The limit lim as x goes to inf of x^x has an indeterminate form.

False

Using partial fractions, we can write: x / (x²+2x+1)(x+2) = Ax+B / x²+2x+1 + C / x+2

False

To evaluate the integral of x² cos(x) dx using integration by parts, if we apply the decision process taught in class, we would set u = cos(x) and dv = x² dx.

False

The integral from -1 to 1 of 1/x dx diverges.

True

The sequence {cos(npi)} diverges.

True

All unbounded sequences are divergent.

True

The series Sigma from n=0 to inf of (-2/3)^n sums to 3.

False

Suppose an > 0 for all n and lim as n goes to inf of an = 0. Then Sigma from n=1 to inf of an converges

False

If an alternating series converges conditionally, then it also converges absolutely.

False

The series Sigma from n=0 to inf of (-1)^n / n+1 sums to 0

False

The power series of Sigma from n=0 to inf of (2-3x)^n / 7^n is centered at x=2/3

True

If Sigma from n=0 to inf of anx^n is a power series and lim as n goes to inf of (an)^1/n = inf, then the series converges for all real numbers.

False

The MacLaurin series for the function f(x) = 1 / (1-x)² is given by Sigma from n=1 to inf of nx^n-1

True