DERIVATIVES UNIT PROBLEM SETS

PROBLEM SET #1 – Rate of Change ***Calculators Not Allowed***

Find the average rate of change for each function between the given values:

NJCTL.org

NJCTL.org

  1. from to
  1. from to
  1. from to
  1. from to
  1. from and
  1. from to
  1. from to
  1. fromand
  1. from to
  1. from to
  1. Andrew is a physics student testing the rate of change of objects he can throw. Given his calculations, if he throws the baseball from the top of a hill, it follows the equation . He wants to know the average rate of change of the ball for each of the following time periods: (Calculator allowed)

a)

b)

c)

NJCTL.org

PROBLEM SET #2 –Slope of a Curve ***Calculators Not Allowed***

For problems #1-8, find the limit of the function at the given point:

  1. a) Find the derivative of the function .

b) What is the value of the derivative at ?

  1. a) Find if the function is .

b) What is the value at ?

  1. a) Find the derivative of .

b) What is the value of ?

  1. a) Find of the function

b) What is the slope at ?

  1. a) Find if .

b) What is the slope at ?

  1. a) Find if .

b) What is the slope at ?

  1. a) Find the derivative of .

b) What is the slope at ?

  1. Of the functions you have worked with, which type of functions have the same average rate of change as their instantaneous rate of change? (i.e. ) Try taking the average rate of changes of the examples above.

a) Linear

b) Quadratic

c) Cubic

d) Square Root

PROBLEM SET #3 –Derivative Rules ***Calculators Not Allowed***

  1. Find the derivative of the function .
  1. Find if the function is .
  1. a) Find the derivative of .

b) What is the value of ?

  1. Find for the function .
  1. a) Find if .

b) What is the slope at ?

  1. a) Find if .

b) What is the slope at ?

  1. a) Find the derivative of .

b) What is the slope at ?

  1. a) Find if .

b) What is the slope at ?

  1. a) Find if .

b) What is the slope at ?

PROBLEM SET #4 –Higher Order Derivatives ***Calculators Not Allowed***

  1. Find the 1st and 2ndderivative of the function .
  1. Find and of the function.
  1. Find the 1st and 2nd derivative of .
  1. Find and of the function .
  1. Find if
  1. Find if .
  1. Find the derivative of .
  1. Find the 1st and 2nd derivatives of

a) where

b) where

c) generically for all

  1. Looking back at problem 8c above, we begin to see a pattern with derivatives. As we take derivatives, our exponents are used as coefficients and multiplied together. If we were to continue taking derivatives until the exponent is zero, we can see that the last coefficient is equal to the factorial () of the original power (any derivative after that will equal zero). If we stopped at any other time along that path, we would have a permutation of the exponents equal to where “a” is the original exponent and “d” is the specific derivative. Using this method, find the following derivatives: (calculator OK)

a) 8th derivative of

b) 10th derivative of

PROBLEM SET #5 –Trigonometry Rules ***Calculators Not Allowed***

  1. a) Find the derivative of the function .

b) What is the value of the derivative at ?

c) What is the value of the derivative at ?

  1. a) Find if the function is .

b) What is the value of the derivative at ?

c) What is the value of the derivative at?

  1. a) Find the derivative of .

b) What is the value of ?

  1. a) Find of the function .

b) What is the derivative at ?

  1. a) Find if .

b) What is the slope at ?

  1. Find the first four derivatives of .
  1. Find the first four derivatives of
  1. Find if .
  1. a) Find the derivative of .

b) What is the value of the derivative at .

PROBLEM SET #6 –Product/Quotient Rule ***Calculators Not Allowed***

  1. Find if the functionis .
  1. Find of the function is .
  1. Find the derivative of .
  1. Find if .
  1. Find if .
  1. Find if .
  1. Find the derivative of (without using trig shortcuts).
  1. Find if .
  1. Find if .
  1. Using any prior rules, find the 1st and 2nd derivatives of .
  1. **Show using product rule that the derivative of is .

PROBLEM SET #7 Derivatives Using Tables ***Calculators Allowed***

x / f(x) / f’(x) / g(x) / g’(x)
-2 / -19 / 16 / -11 / 19
-1 / -6 / 10 / -2 / 4
0 / 1 / 4 / -1 / -1
1 / 2 / -2 / -2 / 6
2 / -3 / -8 / 1 / 7
  1. Givenfind:

find:

  1. Given find:

find:

  1. Givenfind:

find:

  1. Givenfind:

find:

t / g(t) / g’(t) / h(t) / h’(t)
0 / -10 / 6 / -1 / -9
1 / -3 / 1 / -3 / 5
2 / 4 / 14 / -1 / -1/4
3 / 2 / -1.5 / -2 / 5
4 / 0 / 8 / 1/2 / 1
  1. Givenfind:

find:

  1. Givenfind:

find:

  1. Givenfind:

find:

  1. Given find: 1)

find:

PROBLEM SET #8 –Tangent/Normal Lines ***Calculators Not Allowed***

For each question find the equations of the tangentnormal lines at the given value.

  1. at
  1. at
  1. at
  1. at
  1. at
  1. at
  1. at
  1. at

PROBLEM SET #9 –Derivatives of Log & e ***Calculators Not Allowed***

  1. Find the derivative of:
  1. Find for
  1. Differentiate:
  1. Find the derivative of:
  1. Find for:
  1. Find for:
  1. Find the derivative of:
  1. Find for:
  1. Differentiate:
  1. Find the derivative of :

PROBLEM SET #10 –Chain Rule ***Calculators Not Allowed***

  1. Find for :
  1. Given: Find
  1. Given: Find
  1. Find for :
  1. Differentiate:
  1. Given: Find
  1. Find if :
  1. Differentiate:
  1. Given: Find
  1. Differentiate:
  1. Find if
  1. Find for :
  1. Given: Find
  1. Find if
  1. Differentiate:

PROBLEM SET #11 –Derivs. of Inv. Functions ***Calculators Not Allowed***

  1. If and , find
  1. Find the derivative of the inverse of
  1. If and find
  1. If and find
  1. If and find
  1. Find the derivative of the inverse of
  1. If and find
  1. If and Find
  1. If Find
  1. If and Find

PROBLEM SET #12 –Contin. vs. Differentiability ***Calculators Not Allowed***

  1. If a function is differentiable on a given interval, it is also continuous.

a. True

b. False

  1. For a function to be differentiable, it: (choose all that apply)

a. Must have no discontinuities

b. Can have discontinuities

c. Must have no vertical tangent lines

d. Can have vertical tangent lines

e. Must not have corners

f. May have cusps

  1. At which values of x is not differentiable?
  1. At which values of x is not differentiable?
  1. At which values of x is not differentiable?

PROBLEM SET #13 –Der. of Piecewise & Abs. Value ***Calcs. Not Allowed***

  1. a. What is the equivalent piecewise function for the following?

b. What is the derivative?

  1. a. What is the equivalent piecewise function for the following?

b. What is the derivative?

  1. Find the derivative of the following function:
  1. Find the derivative of the following function:
  1. What values of and will make the function differentiable over the interval ?
  1. What values of and will make the function differentiable over the interval ?
  1. What values of and will make the function differentiable over the interval ?

For questions 8 – 10, choose all that apply:

a. is continuous at

b. is differentiable at

c. is not continuous at

d. is not differentiable at

a. is continuous at

b. is differentiable at

c. is not continuous at

d. is not differentiable at

a. is continuous at

b. is differentiable at

c. is not continuous at

d. is not differentiable at

PROBLEM SET #14 –Implicit Differentiation ***Calculators Not Allowed***

  1. Find: +
  1. Find :
  1. Find :
  1. Find :
  1. Find :
  1. Find :
  1. Find :
  1. Find :
  1. Find :
  1. Find :
  1. Find :
  1. Find the slope of the tangent line at the point (1,2) for:
  1. Find the slope of the tangent line at the point (16,-1) for:
  1. Find the slope of the tangent line at the point (1,1) for:
  1. Find the equation of the tangent line at the point (4,8) for:
  1. Find the equation of the tangent line at the point (1,0) for:
  1. Find the equation of the tangent line at the point (-3,3) for:

Limits and Continuity- Answer Keys

NJCTL.org

Problem Set #1 – Rate of Change

  1. 5
  2. 4
  3. a.

b.

c.

Problem Set #2– Slope of a Curve

  1. a.

b.

  1. a.

b.

  1. a.

b.

  1. a.

b.

  1. a.

b.

  1. a.

b.

  1. a.

b.

  1. a. linear

Problem Set #3 – Derivative Rules

  1. a.

b.

  1. a.

b.

  1. a.

b.

  1. a.

b.

  1. a.

b.

  1. a.

b.

Problem Set #4 – Higher Order Derivatives

  1. a.

b.

c.

  1. a.

b.

Problem Set #5 – Trigonometry Rules

  1. a.

b.

  1. a.

b.

  1. a.

b.

  1. a.

b.

  1. a.

b.

  1. a.

b.

Problem Set #6– Product/Quotient Rule

  1. or
  2. or
  1. must show work

Problem Set #7 – Derivative Tables

Problem Set #8 – Tangent/Normal Lines

  1. tangent: or

normal: = or

  1. tangent: or

normal: or

  1. tangent: or

normal: or

  1. tangent: or

normal or

  1. tangent:

normal:

  1. at :

tangent:

normal: or

at :

tangent:

normal: or

  1. tangent:

normal:

  1. tangent: or

normal: or

Problem Set #9 – Derivatives of Logs and e

Problem Set #10 – Chain Rule

  1. or (45
  2. 16
  3. −4

Problem Set #11 – Der. of Inverse Fns.

  1. or

Problem Set #12 – Continuity vs. Diff.

  1. True
  2. a, c, e

Problem Set #13 – Derivatives of Piecewise & Absolute Value Functions

  1. a, d
  2. c, d
  3. a, b

Problem Set #14 – Implicit Differentiation

  1. or

NJCTL.org