Differentiation Questions and Answers

Let g(x) = x² + 3x - 5. Use the limit definition of the derivative to differentiate g. g'(x) =_______ Determine the slope of g at x = - 8. g'(- 8) =_______ Submit All Parts

Let f(x) = ∛2 Determine the derivative of f. f'(x) = ______ Determine the slope of f at x = 2. f'(2) = _______

Let g(x)= -3x² + 5 Use the alternative limit definition of the derivative to determine the derivative (slope) of g at x = -9. g'(- 9) =_______

Consider the graph of f given below. Evaluate the following limit. lim △x->0 f(1+ △x) - f(1)/△x = ________

Let h(y) =- √Y Determine the value(s) of y, if any, for which h has a horizontal tangent line. ○ h has a horizontal tangent line at y = □ ○ h has no horizontal tangent lines.

Let f(x)=9/x² Use the limit definition of the derivative to differentiate f. f'(x)= □ Determine the slope of f at x = 11. f'(11)= □

Hannah invested $32,000 in an account paying an interest rate of 2 ¼ % compounded quarterly. Mila invested $32,000 in an account paying an interest rate of 1⅝% compounded continuously. After 13 years, how much more money would Hannah have in her account than Mila, to the nearest dollar?

Let f(x)=9x² + 4x. Use the limit definition of the derivative to differentiate f. f'(x) = □ Determine the slope of f at x = 9. f'(9) = □

Let f(x) = x² - 5 Determine the equations of two unique tangent lines of f that intersect at (-5, 19). Report solutions using slope-intercept form. y =___ y =___

Let h(x)=√x. Use the limit definition of the derivative to differentiate h. h'(x) = ________ Determine the slope of h at x = 1. h' (1) = ______

The limit definition of the derivative Let g(x) = - 8/x-4 Use the limit definition of the derivative to differentiate g. g'(x) = □ Determine the slope of g at x = -8. g'(- 8) = □

Let f(x) = -8. Use the limit definition of the derivative to differentiate f. f'(x) = ________________________ Determine the slope of f at x = -9. f'(- 9) = ________________

Fit a quadratic function of the form f(t) = c₀ + c₁ t+ c₂ t² to the data points (0,7), (1, 8), (2, –3), (3, −6), using least squares. f(t) =

Part 3: Suppose the field has a length of 6,000 feet. Sally buys fencing to enclose it, but has bought 4 feet too much. Rather than throwing away the extra fencing, Sally enclose a slightly larger field by enlarging each side of the field by the same amount, so the fencing encloses the field and a border: What is the largest animal that could be at point X and be entirely within the border region, touching neither the fence nor the field? a)an ant b)a mouse c)a cat d)a cow e)an elephant Part 4: Give the average rate of change of the perimeter, if the length increases from 6,000 feet to 7,000 feet:_________

42)Let f(x)=x²-5 Determine the equations of two unique tangent lines of f that intersect at (-5, 19). Report solutions using slope-intercept form. y = y =

Let h(x) = 1/4 x² + 5 and f(x) = -1/4 x + d Determine the equation f such that f is perpendicular to h. Report the solution using slope-intercept form.

Let g(x) = (-x⁴ + 3 - 9x³)/x² Determine the derivative of g. g'(x) = _________ Determine the interval(s) on which g is differentiable. Report the solution using interval notation. g is differentiable on __________

Let f(x)=2/x^1/6 Determine the derivative of f. f'(x)= Determine the interval(s) on which f is differentiable. Report the solution using interval notation.

Let h(t) 4t^5 + 5t^4 -6t+ 5t³ Determine the derivative of h. h' (t)= Determine the interval(s) on which h is differentiable. Report the solution using interval notation.

Let h(v) = 2v² (v⁵ + 5) Determine the derivative of h. h'(v) = ______ Determine the slope of h at v = 0. h'(0) = _________

Let h(x) =x^5 Determine the derivative of h. h'(x) = Determine the interval(s) on which h is differentiable. Report the solution using interval notation.

Let h(x) =1/x^12 Determine the derivative of h. h'(x) = Determine the interval(s) on which h is differentiable. Report the solution using interval notation.

maximum and minimum values using Lagrange multipliers. Of all points (x, y, z) that x + 4y + 3z=21, find the one that minimizes (x-1)² + (y-1)² + (z-1)². The given function has a minimum at (____ ____ ______ )which satisfies x + 4y + 3z = 21,

Let g(x)=6/∜x. Determine the derivative of g. g'(x) =[ ].Determine the interval(s) on which g is differentiable. Report the solution using interval notation. g is differentiable on [ ]

Let g(x)=-1/x^4 Determine the derivative of g. g'(x)= Determine the interval(s) on which g is differentiable. Report the solution using interval notation.

Let h(x) = x^4 - 7^3 Determine the derivative of h. h'(x) = Determine the interval(s) on which h is differentiable. Report the solution using interval notation.

Let f(x) = 6x² - 3x + 5 Determine the derivative of f. f'(x) = _________ Determine the interval(s) on which f is differentiable. Report the solution using interval notation. f is differentiable on _______

let h(t)=9t^13 -3t^-13 Determine the derivative of h. h' (t)= Determine the interval(s) on which h is differentiable. Report the solution using interval notation.

Let f(x)=- 6x^11 Determine the derivative of f.f'(x)= Determine the interval(s) on which f is differentiable. Report the solution using interval notation.

Find and simplify each of the following for f(x) = 4x-3. (A) f(x + h) (B) f(x+h)-f(x) (C)(f(x+h)-f(x))/h (Do not factor.)

Let g(x) = 9. Determine the derivative of g. Determine the slope of g at X = 7. g'(7)=

Let h(z) = -5z(⁵/²) - 9z(³/²) + 7z(¹²/¹¹) Determine the derivative of h. h'(z) = ______ Determine the slope of h at z = 1. h' (1) = ______

Let f(w)=(-5w-4w^4 +3)/w^1/3 Determine the derivative of f. f'(w)= Determine the interval(s) on which f is differentiable. Report the solution using interval notation. f is differentiable on_______

Let h(x) = (5 - 2x² - 6x⁴)/x⁴ Determine the derivative of h. h'(x) = ________ Determine the interval(s) on which h is differentiable. Report the solution using interval notation. h is differentiable on ____

let g(x)=1/x⁶. Determine the derivative of g. g'(x) =[? ].Determine the interval(s) on which g is differentiable. Report the solution using interval notation. g is differentiable on[ ?]

Find the relative maximum and minimum values. f(x,y)= e^(6x² +6y² +9) Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The function has a relative maximum value of f(x,y)= at (x,y) =______ (Simplify your answers. Type exact answers. Type an ordered pair in the second answer box.) B. The function has no relative maximum value. Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The function has a relative minimum value of f(x,y) = at (x,y) =_____ (Simplify your answers. Type exact answers. Type an ordered pair in the second answer box.) B. The function has no relative minimum value.

Let g(v)=-v^(12/5) + (2)v^(7/3)-(7)v^(9/8) Determine the derivative of g. g'(v)=[?] Determine the slope of g at v = 1. g'(1)=[?]

Let f(z) = (5z + 7)(-8z +9) a)Determine the derivative of f. f'(z)= b)Determine the slope of f at z = -2. f'(-2)=

find the Taylor series of each function at the indicated number. Give the interval of convergence for each series. f(x)=x² In (1 + x/2) ;x=0

You are constructing a cardboard box with the dimensions 2m by 4m. You then cut equal-size squares from each corner so you may fold the edges. What are the dimensions of the box with the largest volume?

You want to enclose a rectangular region with 450 feet of fencing. Express A(w), the area of the region as a function of its with. A(w)=_________

Find the four second-order partial derivatives for f(x,y)=4x^(7).y^(4)+3x^(5).y^(6). fxx = ___ fyy= ___ fxy = ___ fyx = ___

This problem has two parts. Part 1: In 2069, a virus begins in Boston and spreads in such that the number N of people infected grows exponentially with time t. The table below shows how many days it takes from initial outbreak to have various numbers of cases. N = # of 1 2 3 cases million million million t=# of days 108 157 186 How many days until the Boston virus infects five million people? ______ How many days until there are ten million people infected by the Boston virus?______ Part 2: Enter here (using math notation or by attaching in an image) an explanation of your answers.

Consider the curve parameterized by: x =2t^3/2 - 1 and y = 5t. (a.) Find an equation for the line tangent to the curve at t = 1.

Consider the function f(x) = 2x² + 3x - 4 on the closed interval [-2, 1]. Find the exact value of the slope of the secant line connecting (-2, f(-2)) and (1, f(1)). m =___________ By the Mean Value Theorem, there exists c in (-2, 1) so that m = f'(c). Find all values of such c in (-2, 1). Enter exact values. If there is more than one solution, separate them by a comma. c=___________

Which of the following is not a solution to sin x-cos 2x = 0 on the interval [0, 2π)? (a)π/6 (b)π/2 (c)5π/6 (d)3π/2

Let g(y) = (y-6) (y + 8)/(y-7)(y-4) Determine g'(y) g'(y) =

Let h(x) = f(x) · g(x) and j(x) = f(x)/g(x) x 1 2 3 4 f(x) -5 -3 5 2 f'(x) 4 3 -4 1 g(x) -3 2 1 -4 g'(x) -3 0 -4 1 Use the table above to determine the following derivatives. h' (4) = j'(4) =

Let g(t) = 1/t⁸ Determine the derivative of g. g' (t) = Determine the interval(s) on which g is differentiable. Report the solution using interval notation. g is differentiable on

Let f(x) = (3x + 3)/(-3-7x) Determine the derivative of f. f'(x) = Determine the slope of f at x = - 4. f'(-4)=