MAT 196 Final Concept Checks

(integral from −2−3)𝑓 (𝑥) 𝑑𝑥 = (− integral from 3 2)𝑓 (𝑥) 𝑑𝑥.

FALSE

If (integral from 0 to 2)𝑓 (𝑥) 𝑑𝑥 = 3 and (integral from 0 to 2)𝑔(𝑥) 𝑑𝑥 = 2, then (integral from 0 to 2)𝑓 (𝑔(𝑥)) 𝑑𝑥 = 6.

FALSE

If (integral from 0 to 2)𝑓 (𝑥) 𝑑𝑥 = 3 and (integral from 0 to 2)𝑔(𝑥) 𝑑𝑥 = 2, then (integral from 0 to 2)𝑓 (𝑥)𝑔(𝑥) 𝑑𝑥 = 6.

FALSE

𝑛∑𝑘=1 1/𝑘 + 1 + 𝑛∑𝑘=0 𝑘^2 = 𝑛∑𝑘=0 𝑘^3 + 𝑘2 + 1/𝑘 + 1 .

FALSE

The two-sided limit of f(x) as x-> c exists if and only if the left and right limits of f(x) as x-> c exist.

FALSE The two sided limit exists if and only if the left and right limit exists and are equal.

The limit of f(x) as x approaches c is the value f(c).

FALSE When computing the limit of a function as x-> c, we don't care what happens at x=c.

If 𝑓 (𝑥) − 𝑔(𝑥) = 2 for all 𝑥, then 𝑓 and 𝑔 have the same antiderivative.

FALSE, f and g may not be the same function.

If 𝑓 and 𝑔 are differentiable functions, then 𝑑/𝑑𝑥 (𝑓 (𝑔(𝑥))) = 𝑓 ̈(𝑥)𝑔 ̈(𝑥).

FALSE- According to the chain rule, 𝑑/𝑑𝑥 (𝑓 (𝑔(𝑥))) = 𝑓 ̈(𝑔(𝑥))𝑔 ̈(𝑥).

If 𝑓 ̈(3) = 0 and 𝑓 ̈ ̈(3) = −1, then 𝑓 has a local minimum at 𝑥 = 3.

FALSE- According to the second derivative test, 𝑓 has a local maximum at 𝑥 = 3.

If 𝑓 ̈(1) > 0 and 𝑓 ̈(3) < 0, then 𝑓 has a local maximum at 𝑥 = 2.

FALSE- Assuming 𝑓 ̈ is continuous, 𝑓 ̈(𝑐) = 0 for some 𝑐 ∈ (1, 3), but this 𝑐 need not be 2. If 𝑓 ̈ is not continuous, then there might be no point 𝑐 in the interval (1, 3) such that 𝑓 ̈(𝑐) = 0. For the latter example, let 𝑓 (𝑥) = − |𝑥 − 2|.

𝑑/𝑑𝑥 (3𝑥 + 1)^𝑘 = 𝑘(3𝑥 + 1)^𝑘−1.

FALSE- Must simplify or use chain rule and take the inside derivative as well.

The area of the region between 𝑓 (𝑥) = 𝑥 − 4 and 𝑔(𝑥) = −𝑥^2 on the interval [−3, 3] is negative.

FALSE- By definition, the area between the two graphs is always nonnegative. It is equal to (integral from -3 to 3) |𝑓 (𝑥) − 𝑔(𝑥)| 𝑑𝑥.

ln 5 = (integral from 0 to 5)1/𝑡 𝑑𝑡.

FALSE- By the definition of the natural logarithm function as an integral, ln 5 = (integral from 1 to 5) 1/𝑡 𝑑𝑡.

The 𝑛-rectangle lower sum for 𝑓 on [𝑎, 𝑏] is always equal to the corresponding right sum.

FALSE- Consider the function 𝑓 (𝑥) = 𝑥 on the interval [0, 1]. Partition [0, 1] into one single subinterval so that 𝑛 = 1, 𝑥0 = 0, and 𝑥1 = 1. Then the lower sum for 𝑓 is equal to 0, while the right sum is equal to 1. (Sketch the picture.)

If 𝑓 has a critical point at 𝑥 = 0, then 𝑓 has a local minimum or a local maximum at 𝑥 = 0.

FALSE- Critical points are not guaranteed to produce local extrema. For example, consider 𝑓 (𝑥) = 𝑥^3. Then 𝑥 = 0 is a critical point, but not a local extremum.

Denite integrals and indenite integrals are the same.

FALSE- Denite integrals compute signed area under a curve between two specific points and result in a real number. Indefinite integrals give a family of antiderivatives.

If 𝑓 has no local minima on (−∞, ∞), then it will have no global minimum on the interval [1, 4].

FALSE- For example 𝑓 (𝑥) = 𝑥 has no local extrema on (−∞, ∞), but it does attain its global maximum and minimum values on any closed interval. The minimum value of 𝑓 on the interval [1, 4] is 1 at 𝑥 = 1.

If 𝑥 = 1 is the only critical point of 𝑓 and 𝑓 ̈(0) > 0, then 𝑓 ̈(2) must be negative.

FALSE- For example, consider 𝑓 (𝑥) = (𝑥 − 1)^3. Then 𝑥 = 1 is the only critical point of 𝑓 , but both 𝑓 ̈(0) and 𝑓 ̈(2) are positive.

If 𝑓 is continuous and differentiable on [0, 10] and 𝑓 ̈(5) = 0, then there exist points 𝑎 and 𝑏 in (0, 10) such that 𝑓 (𝑎) = 0 and 𝑓 (𝑏) = 0.

FALSE- For example, consider 𝑓 (𝑥) = (𝑥 − 5)^2 + 1. Then 𝑓 ̈(5) = 0, but 𝑓 (𝑥) ≥1for all 𝑥, so it can never be zero.

The left-sum and the right-sum approximations are the same if the number 𝑛 of rectangles is very large.

FALSE- For example, if a function is increasing everywhere, then the right-sum approximations will always be upper approximations, and the left-sum approximations will always be lower approximations, so that the former will always be larger than the latter.

A midpoint sum is always a better approximation than a left sum.

FALSE- For example, if 𝑓 is a constant function, the two are the same (and both give the exact area).

L'Hôpital's rule applies only to limits as 𝑥 →0 or as 𝑥 →∞.

FALSE- For example, it applies to lim 𝑥→1 𝑥−1/𝑥−1 .

If 𝑓 ̈ is continuous on (0, 6) and 𝑓 ̈(3) is positive, then 𝑓 ̈ is positive on all of (0, 6).

FALSE- For example, let 𝑓 (𝑥) = (𝑥 − 2)^2. Then 𝑓 ̈(𝑥) = 2𝑥 − 4 and 𝑓 ̈(3) = 2 > 0,but 𝑓 ̈(1) = −2 < 0 and 1 ∈ (0, 6).

If a limit initially has an indeterminate form, then it can never be computed.

FALSE- For example, lim 𝑥→0 sin 𝑥/𝑥 = 1.

As limit forms, ∞ − ∞ → 0.

FALSE- For example, lim𝑥→∞ (2𝑥 − 𝑥) = ∞.

If a function is transcendental, then so is its derivative.

FALSE- For example, ln 𝑥 has derivative 1/𝑥 which is an algebraic function.

Every local maximum is a global maximum.

FALSE- For example, the function 𝑓 (𝑥) = 𝑥^3 + 6𝑥^2 has a local maximum at 𝑥 = −4 by the first derivative test, but 𝑓 has no global maximum since lim𝑥→∞ 𝑓 (𝑥) = ∞.

If 𝑓 is continuous at 𝑥 = 𝑐, then 𝑓 is differentiable at 𝑥 = 𝑐.

FALSE- For example, 𝑓 (𝑥) = |𝑥| is continuous but not differentiable at 𝑥 = 0.

If lim𝑥→3 𝑓 (𝑥) = 7, then we can make the values of 𝑓 (𝑥) as close to 3 as we like by choosing values of 𝑥 sufficiently close to 7.

FALSE- If lim𝑥→3 𝑓 (𝑥) = 7, then we can make the values of 𝑓 (𝑥) as close to 7 as we like by choosing values of 𝑥 sufficiently close to 3.

If 𝑓 and 𝑔 are differentiable functions, then the derivative of 𝑓 ◦𝑔 is equal to the derivative of 𝑔 ◦𝑓 .

FALSE- In general, 𝑓 ◦𝑔 and 𝑔 ◦𝑓 are different functions.

L'Hôpital's rule can be used to find the limit of any quotient 𝑓 (𝑥)/𝑔(𝑥) as 𝑥 →𝑐.

FALSE- It can be used only when the quotient has indeterminate form 0/0 or∞/∞ .

If a square grows larger, so that its side length increases at a constant rate, then its area will also increase at a constant rate.

FALSE- It follows that the area of the square is increasing faster as the square grows larger.

If a sphere grows larger, so that its radius increases at a constant rate, then its volume will also increase at a constant rate.

FALSE- It follows that the volume of the sphere is increasing faster as it grows larger.

Every algebraic function 𝑓 is continuous at every real number 𝑥 = 𝑐.

FALSE- It is continuous only at numbers in its domain.

The area between any two graphs 𝑓 and 𝑔 on an interval [𝑎, 𝑏] is given by (integral from a to b) (𝑓 (𝑥) − 𝑔(𝑥)) 𝑑𝑥.

FALSE- It is given by the absolute value of this.

(n=10∑𝑘=1 𝑎𝑘)^2= n=10∑𝑘=1 𝑎^2𝑘 .

FALSE- Let 𝑎𝑘 = 1 for all 𝑘. Then the left-hand side is equal to 100, while the right-hand side is equal to 10.

A function 𝑓 is differentiable at 𝑥 = 𝑐 if and only if 𝑓 ̈−(𝑐) and 𝑓 ̈+(𝑐) both exist.

FALSE- Let 𝑓 (𝑥) = |𝑥|. Then 𝑓 ̈−(0) = −1 and 𝑓 ̈+(0) = 1 both exist, but 𝑓 is not differentiable at 𝑥 = 0.

If 𝑓 is continuous on a closed interval [1, 4], then 𝑓 is continuous at every point in [1, 4].

FALSE- See Definition 1.14; 𝑓 need be only right-continuous at 1 and left-continuous at 4 (along with being continuous at every point of the interior).

If 𝑓 is an inverse trigonometric function, then 𝑓 ̈ is also an inverse trigonometric function.

FALSE- See Theorem 2.18.

𝑛∑𝑘=1 (𝑒^𝑥 )^2 = 𝑒^𝑥 (𝑒^𝑥 + 1)(2𝑒^𝑥 + 1)/6

FALSE- Since (𝑒^𝑥 )^2 is a constant that does not depend on 𝑘

𝑑/𝑑𝑥 (integral from a to x^2) sin 𝑡 𝑑𝑡 = sin 𝑥^2.

FALSE- Since the upper limit is a function of 𝑥, it follows from the chain rule. 2𝑥 sin 𝑥^2

Integral 𝑒^𝑥 cos 𝑥 𝑑𝑥 = 𝑒^𝑥 sin 𝑥 + 𝐶.

FALSE- Taking the derivative of the right-hand side we obtain (𝑒^𝑥 sin 𝑥 + 𝐶) ̈ = 𝑒^𝑥 sin 𝑥 + 𝑒^𝑥 cos 𝑥 ≠𝑒^𝑥 cos 𝑥.

Integral sin(𝑥^2) 𝑑𝑥 = − cos(𝑥^2) + 𝐶.

FALSE- Taking the derivative of the right-hand side we obtain− cos(𝑥^2) + 𝐶 ̈ = 2𝑥 sin(𝑥^2) ≠ sin(𝑥^2).

If 𝑓 is continuous on the interval (1, 3), then 𝑓 must have a maximum value and a minimum value on (1, 3).

FALSE- The Extreme Value Theorem does not apply since the interval (1, 3) is not closed. For example, the function 𝑓 (𝑥) = 𝑥 has no maximum or minimum value on (1, 3).

If 𝑓 (−1) = 2 and 𝑓 (5) = 7, then there must exist 𝑐 such that 𝑓 (𝑐) = 4.

FALSE- The Intermediate Value Theorem does not apply since the function is not assumed to be continuous.

The average value of the function 𝑓 (𝑥) = 𝑥^2 − 3 on [2, 6] is𝑓 (6) + 𝑓 (2)/2 = 33 + 1/2 = 17.

FALSE- The average value of the function on an interval is not the average of its values at the endpoints.

The average value of 𝑓 on [1, 5] is equal to the average of the average value of 𝑓 on [1, 2] and the average value of 𝑓 on [2, 5].

FALSE- The average value on one interval is not necessarily the same as the other.

If 𝑓 is a continuous function such that 𝑓 (0) = 𝑓 (9) = 0 and 𝑓 (3) > 0, then 𝑓 is positive on the entire interval (0, 9).

FALSE- The function 𝑓 could have other roots in (0, 9) and thus could change the sign somewhere in the interval.

The Fundamental Theorem of Calculus applies to 𝑓(𝑥) = tan 𝑥 on [0,𝜋].

FALSE- The function 𝑓 is discontinuous at 𝑥 = 𝜋/2.

If 𝑓 ''(2) = 0, then 𝑓 has either a local maximum or a local minimum at 𝑥 = 2.

FALSE- The function 𝑓 will have a local extremum at 𝑥 = 2 only if 𝑓 ̈ changes sign at 𝑥 = 2. One possible counter example to the statement is 𝑓 (𝑥) = (𝑥 −2)^3.

The value of (𝑥 − 𝑐)𝑓(𝑥)/(𝑥 − 𝑐)𝑔(𝑥) at 𝑥 = 𝑐 is equal to the limit of 𝑓(𝑥)/𝑔(𝑥) as 𝑥 approaches c.

FALSE- The given function is not defined at 𝑥 = 𝑐, while the limit might exist.

If 𝑓 (𝑐) = 5, then lim𝑥→𝑐 𝑓 (𝑥) = 5.

FALSE- The limit of a function as 𝑥 → 𝑐 does not depend on the value of the function at 𝑥 = 𝑐.

If limit x->c f(x)=5, then f(c)=5.

FALSE- The limit of a function does not have to be the same as the value of the function. For example, a function might not be defined at 𝑥 = 𝑐 at all.

If 𝑓 ̈ ̈(1) = 0, then 𝑥 = 1 is an inflection point of 𝑓 .

FALSE- The point 𝑥 = 1 is an inflection point only if 𝑓 ̈ ̈ changes sign at 1. For example, 𝑓 (𝑥) = 𝑥 has no inflection points, yet 𝑓 ̈ ̈(𝑥) = 0 so that in particular𝑓 ̈ ̈(1) = 0.

The second derivative test for local extrema always produces the same information as the first derivative test.

FALSE- The second derivative test could be inconclusive, but the first derivative test is always conclusive. For example, let 𝑓 (𝑥) = 𝑥^4. Then the first derivative test tells you 𝑓 has a local minimum at 𝑥 = 0, yet the second derivative test gives no information.

The instantaneous rate of change of a function 𝑓 at a point 𝑥 = 𝑐 can be represented as the slope of a secant line.

FALSE- The slope of a secant line is only an approximation of the instantaneous rate of change. The instantaneous rate of change is the slope of the tangent line.

𝑚∑𝑘=0 √𝑘 + 𝑛∑𝑘=𝑚 √𝑘 = 𝑛∑𝑘=0 √𝑘.

FALSE- The term corresponding to 𝑘 = 𝑚 is counted twice on the left-hand side but only once on the right-hand side.

A function can approach more than one limit as 𝑥 approaches 𝑐.

FALSE- Theorem 1.6, Uniqueness of limits.

If 𝑓 has a global minimum at 𝑥 = 1 on the interval [0, 3], then the global minimum of 𝑓 on (−∞, ∞) must also be at 𝑥 = 1.

FALSE- There are many counterexamples.

If 𝑓 ̈ changes sign at 𝑥 = 3, then 𝑓 ̈(3) = 0.

FALSE- There are many things that can go wrong. For example, 𝑓 might not be differentiable at 𝑥 = 3, i.e., 𝑓 ̈(3) does not exist. A simple example of such function is 𝑓 (𝑥) = |𝑥 − 3|.

If the graph of f has a vertical asymptote at x=3, then limit x->3 f(x)= infinity.

FALSE- There are many things that could go wrong. The limit might actually be -infinity, or the left limit might be infinity while the right limit is -infinity.

If 𝑓 is a function with 𝑓 (2) = 0 and 𝑓 (8) = 0, then there is some 𝑐 ∈ (2, 8) such that 𝑓 ̈(𝑐) = 0.

FALSE- This function never has zero derivative on (2, 8). (It has derivative 1 at every point in (2, 5) and derivative −1 at every point in (5, 8). Rolle's Theorem does not apply to 𝑓 since it's not differentiable at 𝑥 = 5, which is a point in the interval (2, 8).

If 𝑓 is continuous on an interval 𝐼 , then 𝑓 has both a global maximum and a global minimum on 𝐼 .

FALSE- This is true if 𝐼 is a closed interval by the Extreme Value Theorem. In general, this property may fail. For example, 𝑓 (𝑥) = 𝑥 has no global maximum or minimum on (0, 1).

If 𝑓 ̈ ̈(0) does not exist and 𝑥 = 0 is in the domain of 𝑓 , then 𝑥 = 0 is a critical point of the function 𝑓 ̈.

FALSE- To be a critical point of 𝑓 ̈ a point 𝑥 = 𝑐 must be a point in the domain of 𝑓 ̈ and satisfy either 𝑓 ̈ ̈(𝑐) = 0 or 𝑓 ̈ ̈(𝑐) does not exist. Consider 𝑓 (𝑥) = |𝑥|.Then 𝑥 = 0 is in the domain of 𝑓 , but 𝑓 ̈ is not defined at 0, so in particular𝑓 ̈ ̈(0) does not exist. However, 𝑥 = 0 is not in the domain of 𝑓 ̈, so it is not a critical point of 𝑓 ̈.

𝑑/𝑑𝑥 1/𝑥3 = 1/3𝑥^2 .

FALSE- Using power rule to differentiate you get a different answer.

𝑑/𝑑𝑟 (𝑘𝑠 + 𝑟 ) = 𝑘.

FALSE- We are taking the derivative with respect to 𝑟 , so both 𝑘 and 𝑠 are regarded as constants and 𝑑/𝑑𝑟 (𝑘𝑠 + 𝑟 ) = 1.

The function 𝐴(𝑥) = (integral from 0 to x) (3 − 𝑡) 𝑑𝑡 is concave up.

FALSE- We have 𝐴 ̈(𝑥) = 3 − 𝑥 and hence 𝐴 ̈ ̈(𝑥) = −1. Since the second derivative of 𝐴(𝑥) is negative, the function is concave down.

Where a function 𝑓 is positive, its associated slope function 𝑓 ̈ is increasing.

FALSE- When 𝑓 is increasing, the slope function 𝑓 ̈ is positive. A simple counterexample is a linear function with negative slope.

If 𝑦 is an implicit function of 𝑥 and 𝑑𝑦/d𝑥 at x=2 is 0, then the graph of the implicit function has a horizontal tangent line at the point (2, 0).

FALSE- While the graph of the implicit function does have a horizontal tangent line at the point where 𝑥 = 2, the point on the graph need not have 𝑦-coordinate 0.

𝑑/𝑑𝑥 (1/𝑥) = ln 𝑥.

FALSE- d/dx (1/𝑥) = −1/𝑥^2, while 𝑑/𝑑𝑥 (ln 𝑥) = 1/𝑥.

𝑑/𝑑𝑥 (ln |𝑥|) = 1/ |𝑥|.

FALSE- 𝑑/𝑑𝑥 (ln |𝑥|) = 1/𝑥.

𝑑/𝑑v (integral from 2 to u) 𝑓 (v) 𝑑𝑢 = 𝑓 (𝑢).

FALSE- 𝑓 (v) 𝑑𝑢 = [𝑓 (v)𝑢]𝑢2 = 𝑢𝑓 (v) − 2𝑓 (v) = (𝑢 − 2)𝑓 (v)

If 𝑓 (𝑥) = 𝑥^3, then 𝑓 (𝑥 + ℎ) = 𝑥^3 + ℎ.

FALSE- 𝑓 (𝑥 + ℎ) = (𝑥 + ℎ)^3 ≠ 𝑥^3 + ℎ.

𝑓 ̈(𝑥) = 𝑓 (𝑥 + ℎ) − 𝑓 (𝑥)/ℎ .

FALSE- 𝑓 ̈(𝑥) = lim ℎ→0𝑓 (𝑥 + ℎ) − 𝑓 (𝑥)/ℎ .

For every function 𝑓 on [𝑎, 𝑏], the left sum is always a better approximation with 10 rectangles than with 5 rectangles.

False. Consider a function 𝑓 that is constant on [0, 1] except a thin sharp peak around the point 𝑥 = 1/2. Then the left sum with one rectangle is abetter approximation than a left sum with two rectangles since in the latter the second rectangle will be very high. Now copy this picture five times to create a periodic function on [0, 5] which is going to be a counter example for the 5 vs. 10 rectangles scenario.

When using L'Hôpital's rule, you need to apply the quotient rule in the differentiation step.

False. L'Hôpital's rule involves differentiating the numerator and denominator separately.

The function 𝑓(𝑥) = sec 𝑥 is continuous at 𝑥 = 𝜋/2.

False. Since cos (𝜋/2) = 0, sec 𝑥 is not defined at 𝑥 = 𝜋/2.

Integral 1/𝑥^2 + 1 𝑑𝑥 = ln |𝑥^2 + 1| + 𝐶.

False. Taking the derivative of the right-hand side we obtain ln |𝑥^2 + 1| + 𝐶 ̈ = 2𝑥/𝑥2 + 1 ≠ 1/𝑥2 + 1.

The limit of (𝑥 − 𝑐)𝑓(𝑥)/(𝑥 − 𝑐)𝑔(𝑥) as 𝑥 →𝑐 is equal to the limit of 𝑓(𝑥)/𝑔(𝑥) as 𝑥 →𝑐.

TRUE- This follows from the Cancellation Theorem for limits.

If (integral from 0 to 1)𝑓 (𝑥) 𝑑𝑥 = 3 and (integral from -1 to 0)𝑓 (𝑥) 𝑑𝑥 = 4, then (integral from -1 to 1) 𝑓 (𝑥) 𝑑𝑥 = 7.

TRUE

𝑛∑𝑘=0 1/𝑘 + 1 + 𝑛∑𝑘=1 𝑘^2 = 𝑛∑𝑘=0 𝑘^3 + 𝑘2 + 1/𝑘 + 1 .

TRUE

The limit of f(x) as x approaches c might exist even if the value f(c) does not exist.

TRUE The limit of a function as x-> c does not depend on the value of the function at x=c.

If a function is algebraic, then so is its derivative.

TRUE- Algebraic functions are obtained by performing basic arithmetic operations and raising to rational powers. Since the derivative of any power function is again a power function, combining this with the sum, product, and quotient rules for derivatives we see that the derivative of any algebraic function is still an algebraic function.

If 𝑓 is differentiable on (−∞, ∞) and has an extremum at 𝑥 = 2, then 𝑓 ̈(2) = 0.

TRUE- By Theorem 3.3, 𝑥 = 2 is a critical point of 𝑓 , so either 𝑓 ̈(2) = 0 or𝑓 ̈(2) is undefined. Since 𝑓 is assumed to be differentiable everywhere, we must have 𝑓 ̈(2) = 0.

The graph of an implicit function can have vertical tangent lines.

TRUE- Consider 𝑥^2 + 𝑦^2 = 1 and 𝑥 = 1.

If 𝑦 is an implicit function of 𝑥, then there can be more than one 𝑦-value corresponding to a given 𝑥-value.

TRUE- For example, consider 𝑥^2 + 𝑦^2 = 1. Then for 𝑥 = 0 there are two corresponding 𝑦-values: 𝑦 = −1 and 𝑦 = 1.

Suppose an object is moving in a straight path with position function 𝑠(𝑡). If 𝑠(𝑡) is positive and decreasing, then the velocity v(𝑡) is negative.

TRUE- If the position function 𝑠(𝑡) is decreasing, then the object is moving in the negative direction and hence its velocity is negative.

An upper sum approximation for 𝑓 on [𝑎, 𝑏] can never be an under-approximation.

TRUE- In general, it will be an over-approximation, since the upper edge of the rectangle will always lie on or above the graph of 𝑓 .

If a circle grows larger, so that its radius increases at a constant rate, then its circumference will also increase at a constant rate.

TRUE- It follows that the circumference of the circle is increasing at a constant rate which is 2𝜋 times the rate of increase of its radius.

If a square grows larger, so that its side length increases at a constant rate, then its perimeter will also increase at a constant rate.

TRUE- It follows that the perimeter of the square is increasing at a constant rate which is 4 times the rate of increase of the side length.

As limit forms, 2^∞ →∞.

TRUE- Raising 2 to large powers produces larger and larger values, so lim𝑥→∞ 2^𝑥 =∞.

If limit x-> 3= infinity, then the graph of f has a vertical asymptote at x=3.

TRUE- See Definition 1.3.

If lim𝑥→5 𝑓 (𝑥) = ∞, then we can make the values of 𝑓 (𝑥) as large as we like by choosing values of 𝑥 sufficiently close to 5.

TRUE- See Definition 1.9.

If 𝑓 (𝑥) →0+, then 1/𝑓 (𝑥) →∞.

TRUE- See Theorem 1.29.

As limit forms, ∞^2 → ∞.

TRUE- See Theorem 1.34.

If 𝑓 is an exponential function, then 𝑓 ̈ is a constant multiple of 𝑓.

TRUE- See Theorem 2.14.

If 𝑓 ̈ is a constant multiple of 𝑓, then 𝑓 is an exponential function.

TRUE- See Theorem 2.14; the theorem is an "if and only if" statement.

If 𝑓 is a hyperbolic function, then 𝑓 ̈ is also a hyperbolic function.

TRUE- See Theorem 2.20.

If 𝑓 is discontinuous at 𝑥 = 𝑐, then 𝑓 is not differentiable at 𝑥 = 𝑐.

TRUE- See Theorem 2.5 (differentiability implies continuity).

L'Hôpital's rule applies only to limits of indeterminate forms 0/0 or ∞/∞ .

TRUE- See Theorem 3.14. However, quite often other indeterminate forms can be manipulated into one of these forms.

If 𝑓 is both left and right continuous at 𝑥 = 𝑐, then 𝑓 is continuous at 𝑥 = 𝑐.

TRUE- Since lim𝑥→𝑐− 𝑓 (𝑥) = 𝑓 (𝑐) and lim𝑥→𝑐+ 𝑓 (𝑥) = 𝑓 (𝑐) it follows that both one-sided limits exist and are equal. Hence the two-sided limit exists and is equal to the common value of the one-sided limits, i.e., lim𝑥→𝑐 𝑓 (𝑥) = 𝑓 (𝑐). So, 𝑓 is continuous at 𝑥 = 𝑐.

If 𝑓 has a global minimum at 𝑥 = 3 on (−∞, ∞), then the global minimum of𝑓 on the interval [0, 6] must also be at 𝑥 = 3.

TRUE- Since 𝑓 has a global minimum at 𝑥 = 3 on (−∞, ∞), we have 𝑓 (3) ≤ 𝑓 (𝑥) for all in 𝑥 ∈ (−∞, ∞). In particular, 𝑓 (3) ≤ 𝑓 (𝑥) for all in 𝑥 ∈ [0, 6] which implies 𝑓 has a global minimum on the interval [0, 6] at 𝑥 = 3.

If 𝑓 has an inflection point at 𝑥 = 2 and 𝑓 is differentiable at 𝑥 = 2, then 𝑓 ̈has a local extremum at 𝑥 = 2.

TRUE- Since 𝑓 has an inflection point at 𝑥 = 2, the sign of 𝑓 ̈ ̈ changes at 𝑥 = 2. So, according to the first derivative test for local extrema, 𝑓 ̈ has a local maximum or a local minimum at 𝑥 = 2.

If 𝑓 is an even function, 𝑎 is a real number, and 𝑓 is integrable on [−𝑎,𝑎], then (integral from -a,0)𝑓(𝑥) 𝑑𝑥 = (integral from 0 to a)𝑓(𝑥) 𝑑𝑥.

TRUE- Since 𝑓 is an even function, it is symmetric about the 𝑦-axis. It follows that the signed area under 𝑓 between −𝑎 and 0 is simply the mirror image of the signed area under 𝑓 between 0 and 𝑎. Thus the two integrals are equal.

If 𝑓 is an odd function, 𝑎 is a real number, and 𝑓 is integrable on [−𝑎,𝑎], then (integral from -a,a) 𝑓(𝑥) 𝑑𝑥 = 0.

TRUE- Since 𝑓 is an odd function, it is symmetric about the origin.

If 𝑓 is positive and increasing on [𝑎, 𝑏], then any left sum for 𝑓 on [𝑎, 𝑏] will be an under-approximation.

TRUE- Since 𝑓 is positive and increasing, the rectangle dened by the left sum on any subinterval will have its upper left corner touching the graph of𝑓 , and will be completely below 𝑓 since f is increasing. Since this is true for each rectangle, the left sum is an under-approximation. (Sketch the picture.)

Every power function 𝑓(𝑥) = 𝐴𝑥𝑘 is continuous at the point 𝑥 = 2.

TRUE- Since 𝑥 = 2 is in the domain of every power function 𝑓(𝑥) = 𝐴𝑥𝑘, the statement follows from Theorem 1.22.

The average value of 𝑓 on [1, 5] is equal to the average of the average valueof 𝑓 on [1, 3] and the average value of 𝑓 on [3, 5].

TRUE- The average value of 𝑓 on [1, 5] is the same as the average of the two average values.

𝑑/𝑑𝑥 (𝑒^𝜋) = 0.

TRUE- The expression 𝑒^𝜋 is a constant and the derivative of a constant is zero.

The Fundamental Theorem of Calculus applies to 𝑓(𝑥) = sin 𝑥 on [0,𝜋].

TRUE- The function 𝑓 is continuous on the given interval.

If 𝑓 is continuous and differentiable on [−2, 2] with 𝑓 (−2) = 4 and 𝑓 (2) = 0, then there is some 𝑐 ∈ (−2, 2) such that 𝑓 ̈(𝑐) = −1.

TRUE- The function 𝑓 satises conditions of the Mean Value Theorem on the interval [−2, 2].

When a function 𝑓 has a steep slope at a point on its graph, its instantaneous rate of change at that point will have a large magnitude.

TRUE- The instantaneous rate of change is the slope of the function at the point.

The instantaneous rate of change of a function 𝑓 at the point 𝑥 = 1 is given by 𝑓'(1).

TRUE- The instantaneous rate of change of 𝑓 at 𝑥 = 1 is the slope of the tangent line to the graph of 𝑓 at the point where 𝑥 = 1. This slope is exactly what 𝑓 ̈(1) was defined to be.

If 𝑓 is a trigonometric function, then 𝑓 ̈ is also a trigonometric function.

TRUE- This follows from derivative formulas for trigonometric functions.

Every continuous function has an antiderivative.

TRUE- This follows from the Second Fundamental Theorem of Calculus. Youmight not be able to express the antiderivative in terms of elementary functions, but it is always possible to write it in terms of an integral for continuous functions.

If 𝑓 and 𝑔 are differentiable functions, then (𝑓 (𝑥)𝑔(𝑥)) ̈ = 𝑔 ̈(𝑥)𝑓 (𝑥)+𝑓 ̈(𝑥)𝑔(𝑥).

TRUE- This is just another way to express the product rule formula.

𝑑/𝑑𝑠 (𝑘𝑠 + 𝑟 ) = 𝑘.

TRUE- We are taking the derivative with respect to 𝑠, so both 𝑘 and 𝑟 are regarded as constants and 𝑑/𝑑𝑠 (𝑘𝑠 + 𝑟 ) = 𝑘.

Integral(𝑥^3 + 1) 𝑑𝑥 = (𝑥^4/4 + 𝑥 + 2) + 𝐶.

TRUE- that is the correct antiderivative and it is an indefinite integral so the addition of c is needed.