Free example problems + complete solutions for typical linear approximation problems in Calculus, designed to help you learn how to solve them routinely.

**Update:** We now have a *much* better material to help you learn what’s going with linear approximations, including making use of interactive Desmos graphing calculators so you can *see* what it is you’re actually doing. Please visit our Introducing Linear Approximations screen to *really* get this material down for yourself; there are then more practice problems on our Practice Problems: Linear Approximations screen.

It’s all free, and waiting for you! (Why? Just because we’re educators who believe you deserve the chance to develop a better understanding of Calculus for yourself, and so we’re aiming to provide that. We hope you’ll take advantage!)

If you just need practice with linear approximation problems for now, previous students have found what’s below super-helpful. And if you have questions, please ask on our Forum! It’s also free for your use.

PROBLEM SOLVING STRATEGY: Approximations

Question 1: Square roots

Without using a calculator, estimate:**(a)** $\sqrt{4.04}$**(b)** $\sqrt{3.96}$

Question 2: sin

Without using a calculator, estimate:**(a)** $\sin(0.1)$**(b)** $\sin(-0.1)$

Question 3: cos(*pi* + 1/100) (based on an actual exam question)

Without using a calculator, estimate to two decimal places: $\cos{(\pi + 1/100)}$.

Without using a calculator, estimate to two decimal places: $\sqrt{99}$

Question 4: sqrt(99) (based on an actual exam question)

Answer the following:**(a)** Estimate $(1.0003)^{100}$.**(b)** Show that $(1 + \Delta x)^n \approx 1 + n \, \Delta x$, if $\Delta x$ is small compared to 1.

Question 5: (1.0003)^100 & (1+x)^n

You're going to add a coat of paint of thickness 0.02 cm to a cube of edge-length 10 cm. Approximately how many cubic centimeters of paint will you use?

(This question could state instead, "Use differentials to estimate how much paint you will use.")

(This question could state instead, "Use differentials to estimate how much paint you will use.")

Question 6: Adding a coat of paint

You increase a circle's radius by 1%. By approximately what percentage does its area change?

(This question could state instead, "Use differentials to estimate the percentage change in area.")

(This question could state instead, "Use differentials to estimate the percentage change in area.")

Question 7: Increasing a circle's radius

In this question we're going to try to make geometric sense of the differentials associated with a circle's area, and a sphere's volume.**(a)** The formula for the area of a circle is $A_{\text{circle}} = \pi r^2$.
(i) Find $\dfrac{dA_{\text{circle}}}{dr}$.
(ii) The result from (i) should look familiar. What does $\dfrac{dA_{\text{circle}}}{dr}$ represent geometrically? *Hint:* Look at the result in the form $dA_{\text{circle}} = \_\_\_dr$.**(b)** The formula for the surface area of a sphere is $V_{\text{sphere}} = \dfrac{4}{3}\pi r^3$.
(i) Find $\dfrac{dV_{\text{sphere}}}{dr}$.
(ii) The result from (i) should look familiar. What does $\dfrac{dV_{\text{sphere}}}{dr}$ represent geometrically? *Hint:* Look at the result in the form $dV_{\text{sphere}} = \_\_\_dr$.

Question 8: Differential of a circle's area, and of a sphere's volume

In this question we're going to try to make geometric sense of the differentials associated with a circle's area, and a sphere's volume.**(a)** The formula for the area of a circle is $A_{\text{circle}} = \pi r^2$.
(i) Find $\dfrac{dA_{\text{circle}}}{dr}$.
(ii) The result from (i) should look familiar. What does $\dfrac{dA_{\text{circle}}}{dr}$ represent geometrically? *Hint:* Look at the result in the form $dA_{\text{circle}} = \_\_\_dr$.**(b)** The formula for the surface area of a sphere is $V_{\text{sphere}} = \dfrac{4}{3}\pi r^3$.
(i) Find $\dfrac{dV_{\text{sphere}}}{dr}$.
(ii) The result from (i) should look familiar. What does $\dfrac{dV_{\text{sphere}}}{dr}$ represent geometrically? *Hint:* Look at the result in the form $dV_{\text{sphere}} = \_\_\_dr$.

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