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Derivatives and their applications: mixed review

Read the idea, work independently, then explain what changed.

TOPIC 01

Derivatives and their applications: mixed review

A mixed assessment: identify the method, justify it and revise your reasoning.

What you will be able to explain

Connect the chapter skills without relying on the order of the exercises.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Foundation#Your turn

Find the derivative of x² at x=t from the limit.

t=20

The difference quotient uses h≠0; differentiability requires a finite common two-sided limit.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Differentiate the specified function and substitute only after differentiating.

Hint 2

Use this intermediate relation.

[(t+h)^2-t^2]/h=2t+h

Worked solution

Differentiate the specified function and substitute only after differentiating.
Apply the stated relation and retain its conditions.
$f'(20)=\lim_{h\to0}(40+h)=40$
Cancel h only while h≠0, then take the limit.

The requested value is 40.

Checks and common pitfalls: Cancel h only while h≠0, then take the limit.

Think first. Reveal a hint when the class is ready.

02 / Foundation#Your turn

Find f′(1) using the product rule.

f(x)=x^2(x+21)

Quotient denominators stay nonzero; logarithms have positive real arguments and trigonometric derivatives use radians.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Differentiate the specified function and substitute only after differentiating.

Hint 2

Use this intermediate relation.

f\prime=2x(x+t)+x^2

Worked solution

Differentiate the specified function and substitute only after differentiating.
Apply the stated relation and retain its conditions.
$f'(1)=2(1+21)+1=45$
Both factors contribute to the derivative.

The requested value is 45.

Checks and common pitfalls: Both factors contribute to the derivative.

Think first. Reveal a hint when the class is ready.

03 / Foundation#Your turn

Find the increasing and decreasing intervals.

f(x)=x^2-44x

A critical point needs a sign or second-derivative check; closed-interval global extrema also require endpoints.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Differentiate the specified function and substitute only after differentiating.

Hint 2

Use this intermediate relation.

f\prime(x)=2(x-t)

Worked solution

Differentiate the specified function and substitute only after differentiating.
Apply the stated relation and retain its conditions.
$x<22:f'<0;\quad x>22:f'>0$
The derivative sign changes from negative to positive.

The requested relation or conclusion is shown below.

(-\infty,22):\searrow;\quad(22,\infty):\nearrow

Checks and common pitfalls: The derivative sign changes from negative to positive.

Reasoning checklist · self / teacher assessment

State a valid definition or model and its assumptions.
Show the intermediate mathematical relations, not only the final claim.
Check exclusions, units or the interpretation of the result.

Think first. Reveal a hint when the class is ready.

04 / Foundation#Your turn

Find the tangent to y=x² at x=t.

t=23

The difference quotient uses h≠0; differentiability requires a finite common two-sided limit.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Differentiate the specified function and substitute only after differentiating.

Hint 2

Use this intermediate relation.

m=f\prime(t),\quad P=(t,t^2)

Worked solution

Differentiate the specified function and substitute only after differentiating.
Apply the stated relation and retain its conditions.
$m=46$
Apply the stated relation and retain its conditions.
$y-529=46(x-23)$
A slope alone is not a line equation; include the point of contact.

The requested relation or conclusion is shown below.

y-529=46(x-23)

Checks and common pitfalls: A slope alone is not a line equation; include the point of contact.

Reasoning checklist · self / teacher assessment

State a valid definition or model and its assumptions.
Show the intermediate mathematical relations, not only the final claim.
Check exclusions, units or the interpretation of the result.

Think first. Reveal a hint when the class is ready.

05 / Foundation#Your turn

Find f′(1).

f(x)=x^3+24x

Quotient denominators stay nonzero; logarithms have positive real arguments and trigonometric derivatives use radians.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Differentiate the specified function and substitute only after differentiating.

Hint 2

Use this intermediate relation.

f\prime(x)=3x^2+t

Worked solution

Differentiate the specified function and substitute only after differentiating.
Apply the stated relation and retain its conditions.
$f'(1)=3+24=27$
Differentiate term by term.

The requested value is 27.

Checks and common pitfalls: Differentiate term by term.

Think first. Reveal a hint when the class is ready.

06 / Foundation#Your turn

Find the maximum of f on [0,t].

f(x)=x^2,\quad t=25

A critical point needs a sign or second-derivative check; closed-interval global extrema also require endpoints.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Differentiate the specified function and substitute only after differentiating.

Hint 2

Use this intermediate relation.

f\prime(x)=2x\ge0

Worked solution

Differentiate the specified function and substitute only after differentiating.
Apply the stated relation and retain its conditions.
$f(0)=0,\quad f(25)=625$
Compare the endpoints of the closed interval.

The requested value is 625.

Checks and common pitfalls: Compare the endpoints of the closed interval.

Think first. Reveal a hint when the class is ready.

07 / Standard#Your turn

Find the tangent to y=x² at x=t.

t=26

The difference quotient uses h≠0; differentiability requires a finite common two-sided limit.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Differentiate the specified function and substitute only after differentiating.

Hint 2

Use this intermediate relation.

m=f\prime(t),\quad P=(t,t^2)

Worked solution

Differentiate the specified function and substitute only after differentiating.
Apply the stated relation and retain its conditions.
$m=52$
Apply the stated relation and retain its conditions.
$y-676=52(x-26)$
A slope alone is not a line equation; include the point of contact.

The requested relation or conclusion is shown below.

y-676=52(x-26)

Checks and common pitfalls: A slope alone is not a line equation; include the point of contact.

Reasoning checklist · self / teacher assessment

State a valid definition or model and its assumptions.
Show the intermediate mathematical relations, not only the final claim.
Check exclusions, units or the interpretation of the result.

Think first. Reveal a hint when the class is ready.

08 / Standard#Your turn

Find f′(0).

f(x)=e^{27x}

Quotient denominators stay nonzero; logarithms have positive real arguments and trigonometric derivatives use radians.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Differentiate the specified function and substitute only after differentiating.

Hint 2

Use this intermediate relation.

f\prime(x)=te^{tx}

Worked solution

Differentiate the specified function and substitute only after differentiating.
Apply the stated relation and retain its conditions.
$f'(0)=27e^0=27$
The natural exponential reproduces itself under differentiation.

The requested value is 27.

Checks and common pitfalls: The natural exponential reproduces itself under differentiation.

Think first. Reveal a hint when the class is ready.

09 / Standard#Your turn

Find the minimum of x+t²/x for x>0.

t=28

A critical point needs a sign or second-derivative check; closed-interval global extrema also require endpoints.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Differentiate the specified function and substitute only after differentiating.

Hint 2

Use this intermediate relation.

f\prime(x)=1-t^2/x^2

Worked solution

Differentiate the specified function and substitute only after differentiating.
Apply the stated relation and retain its conditions.
$f'=0\Rightarrow x=28$
Apply the stated relation and retain its conditions.
$f''=1568/x^3>0$
Apply the stated relation and retain its conditions.
$f(28)=56$
The negative critical solution is outside the positive domain.

The requested value is 56.

Checks and common pitfalls: The negative critical solution is outside the positive domain.

Think first. Reveal a hint when the class is ready.

10 / Standard#Your turn

Find the tangent to y=x² at x=t.

t=29

The difference quotient uses h≠0; differentiability requires a finite common two-sided limit.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Differentiate the specified function and substitute only after differentiating.

Hint 2

Use this intermediate relation.

m=f\prime(t),\quad P=(t,t^2)

Worked solution

Differentiate the specified function and substitute only after differentiating.
Apply the stated relation and retain its conditions.
$m=58$
Apply the stated relation and retain its conditions.
$y-841=58(x-29)$
A slope alone is not a line equation; include the point of contact.

The requested relation or conclusion is shown below.

y-841=58(x-29)

Checks and common pitfalls: A slope alone is not a line equation; include the point of contact.

Reasoning checklist · self / teacher assessment

State a valid definition or model and its assumptions.
Show the intermediate mathematical relations, not only the final claim.
Check exclusions, units or the interpretation of the result.

Think first. Reveal a hint when the class is ready.

11 / Standard#Your turn

Find f′(0).

f(x)=e^{30x}

Quotient denominators stay nonzero; logarithms have positive real arguments and trigonometric derivatives use radians.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Differentiate the specified function and substitute only after differentiating.

Hint 2

Use this intermediate relation.

f\prime(x)=te^{tx}

Worked solution

Differentiate the specified function and substitute only after differentiating.
Apply the stated relation and retain its conditions.
$f'(0)=30e^0=30$
The natural exponential reproduces itself under differentiation.

The requested value is 30.

Checks and common pitfalls: The natural exponential reproduces itself under differentiation.

Think first. Reveal a hint when the class is ready.

12 / Standard#Your turn

Find the minimum of x+t²/x for x>0.

t=31

A critical point needs a sign or second-derivative check; closed-interval global extrema also require endpoints.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Differentiate the specified function and substitute only after differentiating.

Hint 2

Use this intermediate relation.

f\prime(x)=1-t^2/x^2

Worked solution

Differentiate the specified function and substitute only after differentiating.
Apply the stated relation and retain its conditions.
$f'=0\Rightarrow x=31$
Apply the stated relation and retain its conditions.
$f''=1922/x^3>0$
Apply the stated relation and retain its conditions.
$f(31)=62$
The negative critical solution is outside the positive domain.

The requested value is 62.

Checks and common pitfalls: The negative critical solution is outside the positive domain.

Think first. Reveal a hint when the class is ready.

13 / Transfer#Your turn

A tank volume V(h)=3h². Find dV/dh at h=t.

t=32

The difference quotient uses h≠0; differentiability requires a finite common two-sided limit.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Differentiate the specified function and substitute only after differentiating.

Hint 2

Use this intermediate relation.

dV/dh=6h

Worked solution

Differentiate the specified function and substitute only after differentiating.
Apply the stated relation and retain its conditions.
$V'(32)=192$
The derivative is volume change per height change, not necessarily volume change per time.

The requested value is 192.

Checks and common pitfalls: The derivative is volume change per height change, not necessarily volume change per time.

Think first. Reveal a hint when the class is ready.

14 / Transfer#Your turn

Find f′(0), using radians.

f(x)=\sin(33x)+\cos x

Quotient denominators stay nonzero; logarithms have positive real arguments and trigonometric derivatives use radians.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Differentiate the specified function and substitute only after differentiating.

Hint 2

Use this intermediate relation.

f\prime(x)=t\cos(tx)-\sin x

Worked solution

Differentiate the specified function and substitute only after differentiating.
Apply the stated relation and retain its conditions.
$f'(0)=33\cdot1-0=33$
A radian argument is essential for the familiar sine derivative formula.

The requested value is 33.

Checks and common pitfalls: A radian argument is essential for the familiar sine derivative formula.

Think first. Reveal a hint when the class is ready.

15 / Transfer#Your turn

Find the minimum of x+t²/x for x>0.

t=34

A critical point needs a sign or second-derivative check; closed-interval global extrema also require endpoints.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Differentiate the specified function and substitute only after differentiating.

Hint 2

Use this intermediate relation.

f\prime(x)=1-t^2/x^2

Worked solution

Differentiate the specified function and substitute only after differentiating.
Apply the stated relation and retain its conditions.
$f'=0\Rightarrow x=34$
Apply the stated relation and retain its conditions.
$f''=2312/x^3>0$
Apply the stated relation and retain its conditions.
$f(34)=68$
The negative critical solution is outside the positive domain.

The requested value is 68.

Checks and common pitfalls: The negative critical solution is outside the positive domain.

Think first. Reveal a hint when the class is ready.

End-of-lesson check and correction

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Teacher preparation and assessment

Question sequence

Ask students to name the relevant condition before calculating.

Board plan

Compare valid methods and annotate their conditions.

Anticipated thinking

A correct final value may still hide a missing assumption.

Assessment checklist

Check the method, conditions, reasoning and interpretation separately.