Differentiation: Definition & Fundamental Properties unit test

A test on this unit alone, marked as a percentage and a letter grade — for the test your class is actually sitting, rather than for May. Answer everything, then submit once: seeing the answer to question 3 before attempting question 4 makes the final percentage meaningless.

Paper 1 Paper 2 Paper 3

Each paper is built from this unit’s 22 terms and is the same for everyone, so a teacher can assign “Unit 2, Paper 2” and every student sits the identical test. Multiple choice is marked objectively; the written sections you mark yourself against the model answer and rubric.

Suggested time 36 min 33 points0/17 attempted

Difference-quotient definition

Second derivative units

Higher-order derivatives in motion

Logarithmic differentiation

Derivative of an inverse function

The constant subtracted is f(a)

Recognizing a derivative in disguise

Alternate-form definition

Derivative of a sum

Symmetric difference quotient

Which rule governs

Cusp

Short answer 1. Define or explain: Tangent versus normal line

3 pts

Short answer 2. Define or explain: Corner

3 pts

Short answer 3. Define or explain: Product rule

3 pts

Short answer 4. Define or explain: Differentiability implies continuity

3 pts

Free response

9 pts

This course has no free-response prompt tagged to this unit, so one from elsewhere in the course is used. It is still worth writing — the skill transfers.

NO CALCULATOR. The Taylor series for a function f about x = 4 is given by Σ (from n = 1 to ∞) (x − 4)ⁿ⁺¹ / ((n + 1)·3ⁿ) = (x − 4)²/(2·3) + (x − 4)³/(3·3²) + (x − 4)⁴/(4·3³) + ⋯ + (x − 4)ⁿ⁺¹/((n + 1)·3ⁿ) + ⋯ and converges to f(x) on its interval of convergence.

A. Using the ratio test, find the interval of convergence of the Taylor series for f about x = 4. Justify your answer.

B. Find the first three nonzero terms and the general term of the Taylor series for f′, the derivative of f, about x = 4.

C. The Taylor series for f′ described in part B is a geometric series. For all x in the interval of convergence of the Taylor series for f′, show that f′(x) = (x − 4)/(7 − x).

D. It is known that the radius of convergence of the Taylor series for f about x = 4 is the same as the radius of convergence of the Taylor series for f′ about x = 4. Does the Taylor series for f′ described in part B converge to f′(x) = (x − 4)/(7 − x) at x = 8? Give a reason for your answer.