微积分I（标准路径） · free preview

MIT 18.01 Practice Exam 1B

Practice exam for Unit I: differentiation rules, derivatives from the definition and piecewise functions.

1. MIT 18.01 Practice Exam 1B — how this assessment works

This lesson reproduces the Practice Exam “MIT 18.01 Practice Exam 1B” from MIT OpenCourseWare 18.01 Single Variable Calculus (Fall 2006) (https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2006/), licensed under CC BY-NC-SA 4.0. Work through 7 problems in order: type your reasoning in the answer box or write it on the handwriting pad, then submit for AI grading.

How the grading works Every problem is graded by an AI mathematics teacher that reads the official solution shipped with the course. It checks your work criterion by criterion, awards partial credit for correct steps, and returns written feedback. Equivalent mathematics and valid alternative methods are accepted — a different route to the same result can still earn full marks.

If your answer is incomplete, the feedback names exactly which steps are missing, so you can correct them and try again.

2. MIT 18.01 PE1B Problem 1 — Derivatives by the quotient and product rules

Evaluate each of the following:

\frac{d}{dx}.\frac{\sqrt{x}}{1+2x}|_{x=1}

\frac{d}{du}(u\ln 2u)

(simplify your answer)

3. MIT 18.01 PE1B Problem 2 — Chain rule on $\sqrt{1-k\cos^{2}t}$

a) Evaluate

\frac{d}{dt}\sqrt{1-k\cos^{2}t},

where $k$ is constant.

b) Check your answer to part (a) by showing that if $k=1$ , your answer agrees with the derivative calculated by a simpler method.

4. MIT 18.01 PE1B Problem 3 — Derivative of $1/x^{2}$ from the definition

Derive the formula for

\frac{d}{dx}(\frac{1}{x^{2}})

directly from the definition of derivative.

(You will need to transform the difference quotient algebraically before taking the limit.)

5. MIT 18.01 PE1B Problem 4 — Derivative of $\sin^{-1}x$ by implicit differentiation

Derive the formula for

\frac{d}{dx}\sin^{-1}x

by solving $y=\sin^{-1}x$ for $x$ and using implicit differentiation.

(You may use the known values of $D\sin x$ and $D\cos x$ in your derivation. Your answer must be expressed in terms of $x$ .)

6. MIT 18.01 PE1B Problem 5 — Making a piecewise function differentiable

Find all values of the constants $a$ and $b$ for which the function defined by

f(x)=\begin{cases} ax+b, & x>1\\ x^{2}-3x+2, & x\le 1\end{cases}

will be differentiable.

7. MIT 18.01 PE1B Problem 6 — Two limits related to derivatives

Evaluate the following, with enough indications to show you are not just guessing:

\lim_{u\to 0}\frac{\tan 2u}{u}

\lim_{h\to 0}\frac{e^{h}-1}{h}

(relate it to the value of a derivative)

8. MIT 18.01 PE1B Problem 7 — The hawk's shadow

A hawk is pursuing a mouse. We choose a coordinate system so the mouse runs along the $x$ -axis in the negative direction, and the hawk is flying over the $x$ -axis, swooping down along the exponential curve $y=e^{kx}$ , for some positive constant $k$ . The hawk in flight is always aimed directly at the mouse. It is noon at the equator, and the sun is directly overhead.

When the hawk's shadow on the ground is at the point $x_{0}$ , where is the mouse?

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