Practice Specialization in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

Applying a general theorem or formula to a specific case by substituting particular values for the variables or parameters.

What does this general statement say about MY specific situation?

Showing a random 20 of 50 problems.

Example 1

hard

Specialize the derivative product rule

(fg)'=f'g+fg'

f(x)=x, g(x)=\sin x

and write

(x\sin x)'

Example 2

hard

Common mistake check: specializing

a^2-b^2=(a-b)(a+b)

a=0

gives what identity? Why does it hold?

Example 3

easy

The area of a rectangle is

A = lw

. Specialize to a square of side

s

Example 4

hard

De Moivre:

(\cos\theta+i\sin\theta)^n=\cos(n\theta)+i\sin(n\theta)

. Specialize to

\theta=60^\circ, n=3

and identify the result.

Example 5

easy

Specialize the identity

a^2 - b^2 = (a-b)(a+b)

with

a=7, b=3

to compute

7^2 - 3^2

Example 6

medium

Specialize the matrix-vector product

A\mathbf v

with

A=\begin{pmatrix}1&2\\3&4\end{pmatrix}

and

\mathbf v=\begin{pmatrix}1\\1\end{pmatrix}

Example 7

medium

Specialize Heron's formula

A=\sqrt{s(s-a)(s-b)(s-c)}

(with

s=\tfrac{a+b+c}{2}

) to a

3

4

5

triangle. Confirm the area.

Example 8

medium

Specialize the cosine double-angle formula

\cos2\theta=1-2\sin^2\theta

\theta=30^\circ

Example 9

easy

Specialize the sine sum identity

\sin(A+B)=\sin A\cos B+\cos A\sin B

A=B=45^\circ

and simplify.

Example 10

challenge

Specialize the derivative power rule and chain rule to differentiate

f(x)=(3x)^2

two ways; confirm they agree.

Example 11

medium

Specialize

f(x)=ax^2+bx+c

to find its value at

x=0

. What does this reveal about

c

Example 12

easy

The general formula for the sum of a geometric series is

S_n = \frac{a(r^n-1)}{r-1}

. Specialise to

a=1, r=2

and compute

S_5

Example 13

easy

Specialize

(a-b)^2=a^2-2ab+b^2

to expand

(x-4)^2

Example 14

medium

Specialize the AM-GM inequality

\frac{a+b}{2}\ge\sqrt{ab}

a=b

. What happens?

Example 15

medium

Specialize the compound-interest formula

A=P(1+r)^t

P=100, r=0.1, t=2

Example 16

medium

Specialize the geometric series sum

\frac{a(1-r^n)}{1-r}

a=1,r=\frac12,n=3

Example 17

medium

Specialize the dot product formula

\mathbf u\cdot\mathbf v=u_1v_1+u_2v_2+u_3v_3

\mathbf u=(1,2,3), \mathbf v=(4,-1,2)

Example 18

medium

Specialize the integral

\int x^n\,dx=\dfrac{x^{n+1}}{n+1}+C

(for

n\ne -1

) to

n=4

Example 19

medium

Specialize the identity

\sin(2\theta)=2\sin\theta\cos\theta

\theta=45^\circ

Example 20

hard

Specialize the quadratic formula to

a=2, b=4, c=-30

and find the roots.

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Related Concepts

Generalization Edge Cases