Differentiation Questions and Answers

Let f(x) = 8. Use logarithmic differentiation to determine the derivative. f'(x) = ƒ'(1) = Question Help

(10 points) Use the definition of the derivative to show f'(x) = 2x + 2 if f(x) = x² + 2x This f(x+h)-f(x) means the "long way" by substituting into the limit: f'(x) = lim h→0 h

Consider the following transformed function> y = 7 cos(x +45) - 2. (i) Complete the following "old" original table of values for the parent function. Yold Xold (ii) Write the transformation equations> xnew=? and ynew=? (iii) Complete the "new" transformed table of values for the new sinusoidal function. Xnew Ynew

y = (7x + cot(x)) 5 dy dx Find dy da Type sin(x) for sin(x), cos(x) for cos(x), and so on. Use x^2 to square x, x^3 to cube x, and so on. Use (sin(x))^2 to square sin(x).

Let f(x) = (In x) 4 f'(x) = f'(e¹) =

Consider the differential equation =y-t. dy Is each of the following a solution to y - t? Select true or false for each proposed solution. y=t+2 [Select] y=t+1 [Select] y=t+1+2et [Select]

Find an equation of the tangent line to the graph of fat the given point. f(x) = √√x, (9, 3) y =

8) y = ( √ / + 6) (3x −¹+4) x differentiate the function -2

Find the derivative of f(x) = 3√x - 9 a 10/2 x8. Type your answer without fractional or negative exponents. Use sqrt(x) for √. ƒ'(x) =

Suppose that the position of a particle is given by s = f(t) = 2t³ + 6t+ 9. (a) Find the velocity at time t. v(t) = (b) Find the velocity at time t = 3 seconds. m S a(t) = m S (c) Find the acceleration at time t. m 82 (d) Find the acceleration at time t = 3 seconds. m 8²

Find the derivative of f(x) = = - 4√x 6 x4 Type your answer without fractional or negative exponents. Use sqrt(x) for √. f'(x) =

This exercise uses the radioactive decay model. The half-life of radium-226 is 1600 years. Suppose we have a 23-mg sample. (a) Find a function m(t)= mo2-/h that models the mass remaining after t years. m(t)= 23 2 1 1600 (b) Find a function m(t) = moet that models the mass remaining after t years. (Round your r value to six decimal places.) m(t) = x (c) How much of the sample will remain after 2000 years? (Round your answer to one decimal place.) mg (d) After how many years will only 19 mg of the sample remain? (Round your answer to one decimal place.) vr

Use the product rule to find the derivative of (-3x¹0 + 5x³)(7e² - 7) Use e^x for et. You do not need to expand out your answer.

Simplify the expression (3i + 2)(1 - i). O-i-1 O i +5 Oi-12

Find the accumulated present value of an investment over a 7 year period if there is a continuous money flow of $5,000 per year and the interest rate is 1.2% compounded continuously.

If f(x) = 3x² − 7x + 6, find ƒ'(1). - Use this to find the equation of the tangent line to the parabola y.= 3x² - 7x + 6 at the point (1, 2). The equation of this tangent line can be written in the form y = mx + b where m is: and where b is:

If f(x) f'(x) = = ƒ'(4) = 5 sin x 2 + cos x then

Let f(x) = 2x + 14 - 3e. Then the equation of the tangent line to the graph of f(x) at the point (0, 11) is given by y = mx + b for m = b=

Use the quotient rule to find the derivative of 5e + 6 4x¹0 - 6x³ Use e^x for e. You do not need to expand out your answer. Be careful with parentheses!

Find the vertical asymptotes of the rational function y = 2-16 X Ox= 0 and x = 1 Ox= 1 and x = -1 Ox=4 and x = -4 y = 4 and y = -4

Let f(x) x + 8 . Find the values of a where f'(x) = 7. Give exact answers (not decimal approximations). The greater solution is x = The lesser solution is =

If g(x) = 1 - 2x + 3x2, find the average rate of change of the function as x varies from 2 to 5. 19 25 33 60

If f(x) ƒ'(x)= - 3x sin x cosa, find Find f'(4) =

y = (4x³ + tan(x)) ³ dy dx Find dy dx = Type sin(x) for sin(x), cos(x) for cos(x), and so on. Use x^2 to square x, x^3 to cube x, and so on. Use (sin(x))^2 to square sin(x).

If f(x) = f'(x) = ƒ' (2) = 2 sin x I 3 + cos x then

Radio stations in a certain country use a sequence of 5 or 6 letters as their station identification call letters. The first letter must be W, N, C, Q, or R. Assume there are no restrictions on the remaining letters, and repetition is allowed. a) How many 5-letter station identifications are possible? b) How many 6-letter station identifications are possible? c) How many total station identifications are possible? d) The identification for a randomly-chosen radio station is 5 letters in length. What is the probability that all five letters are different? a) Set up the expression that would be used to calculate the number of possible 5-letter station identifications. The expression is (Do not simplify.) There are possible 5-letter station identifications. (Simplify your answer.) b) There are possible 6-letter station identifications. (Simplify your answer.) C c) The total number of possible station identifications is (Simplify your answer.) d) If the identification for a randomly-chosen radio station is 5 letters in length, then the probability that all five letters are different is (Round to three decimal places as needed.) 4

(10 points) If f(3) = 8, f'(3) = 4, g(3) = 2, g'(3)=-6, find F'(3) if F(x)=√f(x) g(x)

Let f(x) = 4x + 14 if x < -2 √x+38 if x > - 2 if x = - 2 2 lim H12 Select all statements below that you agree with. Note: You may be checking more than one box. No partial credit. Of(-2) is defined. lim HI12 f(x) exists. f(x) = f(-2). # O The function is continuous at x = -2. The function is not continuous at x = -2.

Use the product rule to find the derivative of (10x5 + 2x)(8 + 9) Use e^x for e. You do not need to expand out your answer.

Let u(x) = sin(x) and v(x) = x² and f(x) u'(x) = v'(x) = f' = u'v - uv' v² = = u(x) v(x)

Find the derivative of the function g(x) = (5x² − x - 1) e g'(x) = -

If f(x) f'(x) = f'(2) = 5x(sin x + cos z), find

Determine O y = log; (x² + 1) dy dx (T²+1) In 3 21 (T²+1) (7²+1) In 3 20 (T²+1) 21 for the following function:

(1) Which is the correct formula for finding the derivative of the product of two functions? (fg)' = f' + g' O(fg)' = fg' + gf' (fg)' = f'g' (2) Use the correct formula above to find the derivative of the function f(x) = x³e².

Let f(x) Ho MINIMA Then f'(x) b- | 9 sin r 2 sin x + 6 cos x The equation of the tangent line to y = f(x) at x = 0 can be written in the form y = = mx + b where and

Given the function: f(x) 1.) (a) Find: f(-4) 11 - 3x²5x + 2, find the following functional values. (b) Find: f(3)

If f(x) ƒ'(x)= = 5x sin x cos x, find Find ƒ'(1) =

Find the derivative of: f(x) = x(1 - x)² O (1 - x)(1 - 3x) 2(1-x) O (1 - x)(-1 - x) O-2 (1-x)

Let f(x) = 9 cos x + 3 tan x ƒ'(x) = - f' (57) 3 =

Determine dy dx at x = -2 for y = 3u² + 2u and u = √x² + 5.

If f(x) f'(x) = - f'(2) = 4 sin x 2 + cos x 7 then

y = xe 2x 0 ≤ x ≤ In3 On this interval, the maximum value(s) occur at x = In3 1 2 0 O two of the above are correct Shy

Find the derivative of f(x) = 7√x 3 x10 # Type your answer without fractional or negative exponents. Use sqrt(x) for √. ƒ'(x) =

Differentiate y = O 0 O O 2√x 4x³ 2√x +1 A√ √x + √2 1 2√x + √√/x √x + √x

In(x-3) < 0 if and only if Ox< 4 Ox> 3 Ox>4 03<x< 4 00<x< 0 < x < 4

If f(x)= ex, which of the following lines is an asymptote to the graph of f? Oy=x Oy=0 Oy = -x Oy=1 O x = 0 47

y = (6x³ + 3) ³y = (6x³ + 3) 5 Let y = Find dy dx dy dy dx dx dy da Type sin(x) for sin(x)sin(x), cos(x) for cos(x) cos(x), and so on. Use x^2 to square x, x^3 to cube x, and so on. Use (sin(x))^2 to square sin(x). Do NOT simplify your answer.

A cell culture contains 2 thousand cells, and is growing at a rate of r(t) = 6e0.00t thousand cells per hour. Find the total cell count after 3 hours. Give your answer accurate to at least 2 decimal places.

y = Find dy dx (2x² + 9e²) ² dy dx Type sin(x) for sin(x), cos(x) for cos(x), and so on. Use x^2 to square x, x^3 to cube x, and so on. Use (sin(x))^2 to square sin(x). A

Find the derivative of the function g(x) = g'(x) = e¹ 5 - 4x