Derivatives

Basic Derivatives

100

What is the derivative of y = 3x² ?

y' = 6x

200

Find the derivative of f(x) = 5

0

300

Find the derivative of f(x) = 6x³ + x - 2

f'(x)=18x²+ 1

400

Find the derivative of f(x) = 3sinx - 2cosx

f'(x)=3cosx + 2sinx

500

What is the equation of the tangent line to 13x⁴ + 3x² - 6x +3 at x=1?

y-52=13(x-1) or y=13x+39

Product and Quotient Rule

100

Find the derivative of y = 3x² ln(x)

y'=3x+ 6x ln(x)

200

f(x) = (3x - 2x²)(5 + 4x)

f'(x)=-24x² + 4x + 15

300

Find the derivative of y = (5x - 2)/(x² + 1)

y'=(-5x2 + 4x + 5)/(x2 + 1)²

400

Find the derivative of f(x) = (2x + 5)/(x)^1/2

f'(x)=(2x - 5)/(2x^3/2)

500

f(x)=x²sinx, what is f′(x)?

2xsinx+ x²cosx

Grab Bag

100

Derivative of y=csc (x) at x=pi/4

-sqrt(2)/4

200

Let f be a function such that (f(2+h)-f(2))/h=5. Which of the following must be true? Choose all that apply.

I. f is continuous at x=2

II. f is differentiable at x=2

III. The derivative of f is continuous at x=2

I, II, and III

300

lim_h->0(e^-1-h-e^-1)/h=

-1/e

400

Solve a and b in order for g(x) to be continuous and differentiable at x=0.   What is a+b?

          <   ax+b   when x>0

g(x)=  <

           <  1-x+x²   when x<=0

0

500

f(x)=2g(x)+g(x)/h(x)

g(2)=3  g'(2)=-2

h(2)=-1  h'(2)=4

Find the equation of the tangent line in slope intercept form when x=2.

y=10x-17

Taylor Polynomials

100

What is the coefficient of (x-2)³ in the Taylor series of lnx centered at x=2?

1/4

200

Find a second degree Maclaurin polynomial for h(x)=7xe^x

y=7x+7x²

300

Suppose that g is a function such that g(5)=3, g'(5)=-2, g''(5)=7, and g'''(5)=-3. 

Write a third degree Taylor Polynomial centered at x=5 and use this to approximate g(4.9). 

3.236

400

The Maclaurin series for f(x) is f(x)=1+x/2+(x^2)/6+(x^3)/24+....

f''(0)=?

1/3

500

x  f(x)  f'(x)  f''(x)  g(x)  g'(x)

3   1     -2      7       4       -5

For n>=2, the nth derivative of g at x=3 is given by f⁽ⁿ⁾(3)=f^(n-2)(3). Find the third degree Taylor polynomial for g(x) about x=3. 

4-5(x-3)+(x-3)²/2-(x-3)³/3

Error

100

Suppose a function f is approximated with a fourth-degree Taylor polynomial about x=1. If the maximum value of the fifth derivative between x=1 and x=3 is 0.01. Then the maximum error incurred using this approximation is... (give 5 decimals)

0.00267

200

How accurate would a fourth-degree Taylor polynomial for cos x about x=0 be to approximate the value of cos(0.8)? State the maximum error to 3 decimal places. 

0.003

300

Freebie! Pick a number 1-10. If you get the correct number, you get the 300 points for this question

2

400

Write a Maclaurin polynomial for function f that has derivatives of all orders on the interval (-1,1). Assume f(0)=1, f'(0)=-1/4, f''(0)=3/8, and the |f⁽⁴⁾(x)|<=6 for all x in the interval (-1,1). What is the maximum possible error for the approximation of f(0.5)? 

(Round to 3 decimal places)

0.016

500

The Taylor series about x=3 for a certain function f converges to f(x) for all x in the interval of convergence. The nth derivative of f at x=3 is given by

f⁽ⁿ⁾⁼(-1)ⁿ*n!/(5ⁿ(n+3)) and f(3)=1/3

What is the least degree polynomial needed to approximate f(4) with an error less than 1/4000.

Third degree