Phase 2 Exam

Single-Variable Calculus - Phase Exam

20 interleaved questions drawn from every lesson in this phase. Interleaving mixes topics on purpose - that difficulty is what builds durable, transferable understanding. Aim for 70%+ before advancing; below that, revisit the flagged lessons.

How this exam teaches

Questions are shuffled across lessons (not blocked by topic) so you practice choosing the right idea. Missed questions are added to your review queue automatically.

Q1 Easy

For lim_{x→c} f(x)=L, the condition 0<|x−c|<δ explicitly excludes x=c because the limit:

requires f(c) to exist

describes the approach to c, independent of the value at c

is always equal to f(c)

only makes sense for continuous f

Q2 Easy

The derivative f′(x) is defined as the limit of:

f(x+h)−f(x)

(f(x+h)−f(x))/h as h→0

(f(x+h)−f(x))/h as x→0

f(x)/x

Q3 Easy

The derivative of sin(x²) is:

cos(x²)

2x cos(x²)

2x sin(x²)

cos(2x)

Q4 Easy

The linear (first-order Taylor) approximation of f about a is:

f(a)

f(a)+f′(a)(x−a)

f′(a)(x−a)

f(a)+f″(a)(x−a)²/2

Q5 Easy

At an interior local maximum of a differentiable function, Fermat’s theorem guarantees:

f″=0

f′=0

f=0

f is increasing

Q6 Easy

FTC Part 1 states that d/dx ∫_a^x f(t)dt equals:

F(x)−F(a)

f(x)

f′(x)

Q7 Easy

The geometric series ∑ ar^n converges if and only if:

a≠0

|r|<1

r>0

a<1

Q8 Medium

f(x)=(x²−9)/(x−3) at x=3 has:

a jump discontinuity

a removable discontinuity with limit 6

no limit

an infinite discontinuity

Q9 Medium

Which is TRUE about the relationship between continuity and differentiability?

continuous ⇒ differentiable

differentiable ⇒ continuous

they are equivalent

neither implies the other

Q10 Medium

Differentiating x²+y²=25 implicitly gives 2x+2yy′=0, so y′ equals:

−x/y

x/y

−y/x

−2x

Q11 Medium

In the Maclaurin series e^x=∑ x^k/k!, the coefficient of x³ is:

1/3

1/6

Q12 Medium

If f′(c)=0 and f″(c)>0, then c is a:

local maximum

local minimum

saddle point

global maximum

Q13 Medium

∫₀¹ (4x³) dx equals:

x⁴

Q14 Medium

Which statement is correct about the harmonic series ∑1/n?

it converges because 1/n→0

it diverges even though 1/n→0

the ratio test proves it converges

it converges to 1

Q15 Medium

The Intermediate Value Theorem lets you conclude x³+x−1 has a root in (0,1) because it is continuous and:

f(0) and f(1) have opposite signs

f is increasing

f is a polynomial

f(0)=f(1)

Q16 Medium

For a balance under continuous compounding, A(t)=A₀e^{rt}. Its instantaneous growth rate A′(t) equals:

rA₀

rA(t)

A(t)/r

Q17 Medium

A 13 ft ladder slides down a wall; the base moves out. Which relation should you differentiate w.r.t. t for a related-rates problem?

x+y=13

x²+y²=13²

xy=13

x²−y²=13

Q18 Medium

For a bond, ΔP/P ≈ −DΔy + ½CΔy². The convexity term CΔy² is:

the linear risk

the second-order Taylor curvature correction

always negligible

the same as duration

Q19 Medium

A firm maximizes profit π=R−C. The first-order condition is:

R=C

R′=C′ (marginal revenue = marginal cost)

R′=0

C′=0

Q20 Medium

The present value of a continuous stream c(t) discounted at rate r over [0,T] is:

∫₀ᵀ c(t)dt

∫₀ᵀ c(t)e^{−rt}dt

∫₀ᵀ c(t)e^{rt}dt

c(T)e^{−rT}

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