Calculus practice

Read what the question requests before calculating: a function, a rate at a point, a change, or an accumulated total. Correct answers and revealed steps stay visible until you continue.

Read a formal question

Question. Let $f:\mathbb R\to\mathbb R$ satisfy $f(2)=7$ and $f'(2)=-3$. Use a first-order approximation to estimate $f(2.04)$.

Translate the notation

The function takes one real number and returns one real number. At input 2, its output is 7 and its rate of change is −3. “First-order approximation” means use the local straight-line prediction.

Work through the calculation

The input change is $h=2.04-2=0.04$. The predicted output change is $f'(2)h=-3(0.04)=-0.12$. Add that to the starting output:

$$f(2.04)\approx7-0.12=6.88.$$

−3 is the rate; −0.12 is the predicted change; 6.88 is the predicted new output. The given information does not determine the exact value or guarantee a particular error bound at this step size.

Practice a particular topic

Derivatives and local predictions

Review derivatives.

Choosing a derivative rule

Review the rules.

Definite integrals and running totals

Review the fundamental theorem.

Partial derivatives

Review partial derivatives.