Calculus I · Unit 2A · diagnostic

Derivative Prerequisite Diagnostic

Prerequisite Diagnostic

Complete without a calculator unless your course normally permits one.

Exercise

Expand $(x+h)^2$ .

Exercise

Factor $x^2-9$ .

Exercise

Simplify $(x^2-4)/(x-2)$ for $x\ne2$ .

Exercise

Rewrite $1/x^3$ with a negative exponent.

Exercise

Rewrite $\sqrt[3]{x^2}$ with a rational exponent.

Exercise

Simplify $\frac{1/(x+h)-1/x}{h}$ .

Exercise

Solve $3x^2-12x+9=0$ .

Exercise

State the slope formula through $(x_1,y_1)$ and $(x_2,y_2)$ .

Exercise

Write the point-slope equation for slope $4$ through $(2,-1)$ .

Exercise

Evaluate $\lim_{h\to0}[(2+h)^2-4]/h$ .

Exercise

State $\lim_{u\to0}\sin u/u$ .

Exercise

For $f(x)=x^2$ , calculate $f(a+h)-f(a)$ .

Answer reveal

Worked solution

Your setup or answer

Write a real attempt before opening the supplied answer.

Method

How to use the result

Missing zero to two items: begin the unit. Missing three to five: review the attached algebra topics while starting the first conceptual lessons. Missing six or more: repair factoring, exponents, rational expressions, and limits first. This is not gatekeeping; it is the shortest route through the material.

After the explanation

Use the section idea

Reading lens

Treat the derivative as a local response rate before treating it as a formula.

Mental model

A function reports an amount; its derivative reports how that amount responds to a small input change.

Decision

Check prerequisites, identify the input and output, and attach units before calculating.

Common trap

Reading derivative notation as an ordinary fraction or skipping the limit meaning.

Check yourself

Can you explain what a derivative value means in one complete sentence with units?

Source & rights

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