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′(fg)'=f'g+fg' to f(x)=x,g(x)=sin⁡xf(x)=x, g(x)=\sin x and write (xsin⁡x)′(x\sin x)'.

Example 2

hard
Common mistake check: specializing a2−b2=(a−b)(a+b)a^2-b^2=(a-b)(a+b) to a=0a=0 gives what identity? Why does it hold?

Example 3

easy
The area of a rectangle is A=lwA = lw. Specialize to a square of side ss.

Example 4

hard
De Moivre: (cos⁡θ+isin⁡θ)n=cos⁡(nθ)+isin⁡(nθ)(\cos\theta+i\sin\theta)^n=\cos(n\theta)+i\sin(n\theta). Specialize to θ=60∘,n=3\theta=60^\circ, n=3 and identify the result.

Example 5

easy
Specialize the identity a2−b2=(a−b)(a+b)a^2 - b^2 = (a-b)(a+b) with a=7,b=3a=7, b=3 to compute 72−327^2 - 3^2.

Example 6

medium
Specialize the matrix-vector product AvA\mathbf v with A=(1234)A=\begin{pmatrix}1&2\\3&4\end{pmatrix} and v=(11)\mathbf v=\begin{pmatrix}1\\1\end{pmatrix}.

Example 7

medium
Specialize Heron's formula A=s(s−a)(s−b)(s−c)A=\sqrt{s(s-a)(s-b)(s-c)} (with s=a+b+c2s=\tfrac{a+b+c}{2}) to a 33-44-55 triangle. Confirm the area.

Example 8

medium
Specialize the cosine double-angle formula cos⁡2θ=1−2sin⁡2θ\cos2\theta=1-2\sin^2\theta to θ=30∘\theta=30^\circ.

Example 9

easy
Specialize the sine sum identity sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin(A+B)=\sin A\cos B+\cos A\sin B to A=B=45∘A=B=45^\circ and simplify.

Example 10

challenge
Specialize the derivative power rule and chain rule to differentiate f(x)=(3x)2f(x)=(3x)^2 two ways; confirm they agree.

Example 11

medium
Specialize f(x)=ax2+bx+cf(x)=ax^2+bx+c to find its value at x=0x=0. What does this reveal about cc?

Example 12

easy
The general formula for the sum of a geometric series is Sn=a(rn−1)r−1S_n = \frac{a(r^n-1)}{r-1}. Specialise to a=1,r=2a=1, r=2 and compute S5S_5.

Example 13

easy
Specialize (a−b)2=a2−2ab+b2(a-b)^2=a^2-2ab+b^2 to expand (x−4)2(x-4)^2.

Example 14

medium
Specialize the AM-GM inequality a+b2≥ab\frac{a+b}{2}\ge\sqrt{ab} to a=ba=b. What happens?

Example 15

medium
Specialize the compound-interest formula A=P(1+r)tA=P(1+r)^t to P=100,r=0.1,t=2P=100, r=0.1, t=2.

Example 16

medium
Specialize the geometric series sum a(1−rn)1−r\frac{a(1-r^n)}{1-r} to a=1,r=12,n=3a=1,r=\frac12,n=3.

Example 17

medium
Specialize the dot product formula u⋅v=u1v1+u2v2+u3v3\mathbf u\cdot\mathbf v=u_1v_1+u_2v_2+u_3v_3 to u=(1,2,3),v=(4,−1,2)\mathbf u=(1,2,3), \mathbf v=(4,-1,2).

Example 18

medium
Specialize the integral ∫xn dx=xn+1n+1+C\int x^n\,dx=\dfrac{x^{n+1}}{n+1}+C (for n≠−1n\ne -1) to n=4n=4.

Example 19

medium
Specialize the identity sin⁡(2θ)=2sin⁡θcos⁡θ\sin(2\theta)=2\sin\theta\cos\theta to θ=45∘\theta=45^\circ.

Example 20

hard
Specialize the quadratic formula to a=2,b=4,c=−30a=2, b=4, c=-30 and find the roots.
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Related Concepts

GeneralizationEdge Cases