The equation for the line tangent to f (x) at the point (7, 11) is given by y = 12x – 9. Use f(7+h) – f(7) this to find lim h→0 h f(7+h) – f(7) lim = 12 h f(7+ h) – f(7) lim h f(7+h) – f(7) | = 11 lim h→0 h f(7+h) – f(7) lim = 7 h

Q: Prove that lim 12 √16x +9y¹2 2 6 4x² +3y does not exist by showing that the limits along different…

A: We have to prove that  limx, y→0, 0 16x4+9y124x2+3y6 does not exist.

Q: If V 16x <f(x)sx+4 for all positive x-values, then what is lim f(x)?

A:

Q: 2(x-1)* - y² Lex-1)* +2y (d) Show lim does not exist.

A: Limit exist

Q: Let (7z 2x2 for I<0, 8z2 + 7x for r20. f(1) = According to the definition of the derivative, to…

A: Given : Derivative any function is as follows let   f(x)= xm

Q: Find the slope of the tangent line to the parabola y=4x-x^2 at the point (1,3) 1) using the…

A: Given

Q: Use the graphs of f and g and their tangent lines at (2, 0) to find lim x) x-2 g(x) y = 1.4(x – 2) y…

A: NOTE: Refresh your page if you can't see any equations. . for the graph f, the equation of the…

Q: 2. Use a table of function values to estimate lim in3x sin 3x sin3x -0.1 0.1 -0.01 b. 0.01 -0.001 c.…

A:

Q: find the point (x,y) at which the graph of y=7x2+5x-6 has a horizontal tangent line.  The function…

A:

Q: Solve for the slope of the functions at the points specified using the limit definition of slope of…

A: Consider the function  f(x) = 12x2− 9x + 15 point (−2, 81).

Q: Assuming h > 0, compute Vx - Vh lim x→h X-h (a); Vh (b). 2h (c) (d) 2h (0)

A:

Q: Q2: (a) By using " L' Hopital rules "; Find the limits of ::- (1) lim,-(7/2)-(tan x)" (2) limy-o…

A:

Q: Which of the following represents the slope of the tangent to f(x)=Vx+1 at x=8 ? Select one: 8+h-/8…

A: As we know that the expression for the slope of the tangent to the curve is:   limh→0fx+h-fxh…

Q: Q1) A- Prove that: cotx = - tan-1x B- Find: 1. lịm 2. lim(v9x2 - x- 3x) cotx

A:

Q: Find the limit. 5xz - 7xy P (1, - 1, 0) x2 + y2 - z2 lim %3D O 5 7/2

A:

Q: Let lim,-0+ f(x) = -2, where 8r - 3 for r <0 f(x): | 4x – 2 for r2 0 f(r) = Then what is (the…

A:

Q: Compute the limit 1 a2 1 - x2 lim xa x а - What can you say about the derivative of f(x) = ?

A: Given limit :

Q: Q4)A) Find the following limits (for two only). x+1 1. limx→-1 x2-x-2 1-Vx 2. limx→1 1-x 2х 3.…

A:

Q: Find the point (xo,yo) on the graph of y = x' + 9x² + 7x – 5, at which the slope of the tangent line…

A:

Q: Find the limit of the function f(x, y) = 4e10z–3y (x, y) → (0,0). as lim (x,y)→(0,0) 4e10x-3y

A:

Q: find the slope of a curve y = x2-2x-2 at the point P(3,1) by finding the limit of the secant slopes…

A: Given yx=x2-2x-2           P3,1

Q: Use the graphs of f and g and their tangent lines at (-1,0) to find f(x) lim Number x-1 g(x) y f(x)…

A:

Q: Use the graphs of f and g and their tangent lines at (2, 0) to find lim f(x) x+2 g(x) y4 y = 1.6(x -…

A: We need to evaluate the value of limx→2f(x)g(x) using the following graph

Q: 2-メ Compite lim %3D Xラム z(T-X)

A:

Q: Consider the function f(x, y) 2y (a) lim (x,y)→(0,0) f (x, y) does not exist by using different…

A:

Q: Find the slope of line drawn tangent to the function. f(x)=x2+7, at the point (-2,11) by using the…

A: Shape of tangent line to the curve is given by first derivative of curve at given periodDerivative…

Q: Show that the tangents to the curve y2 = 4px from any point on the line x = -p are perpendicular.

A: The given curve is,

Q: Q1) A) Find the tangent equation for the function: f(x) = vx2 + 5 at =2 ? B) Find the limits by…

A:

Q: B. Calculate the limits of the following functions.. 1. limx¬-1(2X2 + X + 4) 2. limy--2(y3 + 5y – 1)…

A:

Q: Find the limit. x²y lim (x,y) → (0,0) x² + 8y²

A: limx, y→0, 0x2yx2+8y2

Q: [4] (a) Show that lim (xa)-(0,0) 2x+y? y-mx, or along any parabola y= exists and it's approaches to…

A:

Q: Find a point on the graph of the curve y = (x - 3), where the tangent is parallel to the chord…

A:

Q: B. Find the length of the curve y = x² from x 0 tox = 4.

A:

Q: According to the alternate form of the definition of the derivative, to compute f′(0), we need to…

A:

Q: Compute the equation of the horizontal asymptote of the curve y = (2x- 1)/ (x + 2). y = 2 x = 2 X =…

A: Given; y=2x-1x+2   As we know that; To calculate the horizontal asymptote, solve the limit…

Q: Show that xy lim (x,y)→(0,0) x2 + y8 does not exist.

A: If two different path have two different limit then limit does not exist . Now find the limit along…

Q: The graph of fCx) below To evalvate tue following limits. Ao lim fex) 104 9. B. ling fCX) C.lim f(X)…

A: Given, graph of fxTo find: limx→2-fx limx→2fx limx→4fx limx→5fx limx→∞fx

Q: f'(0) = Is f(1.5)>0? ('Yes on not) lim fCh-)-f(-)

A:

Q: Find the limit: 4z4 – 4y4 lim (1,9) → (3,3) 4yª z² - y?

A: lim(x, y)→(3, 3) 4x4-4y4x2-y2

Q: Suppose that f(x) is a differentiable function (with a continuous derivative), such that f(0)=f'(0)…

A:

Q: Consider the function 2+2+17-2ry+v² if (r, y) # (0,0) f(r,y) = if (1, y) = (0,0). (b) Find…

A:

Q: Find the length of the curve : y = + from x = 1 to x = 3 - 3 4x

A:

Q: Find the point where the tangent to the graph of f(x)= -x^2-6 is parallel to the line y=4x-1

A:

Q: Determine lim In/xy. (x,y)(e4)

A: lim(x,y)-&gt;(e2,4)lnxy

Q: T is discontinuous at -1 because lim f(x) as x--> -1 does not exist. * %23 .7. 5. %23

A:

Q: For f(x) = 2x³ - 2, find ƒ '(x) using the definition f '(x) = lim f(x+h)-f(x) h h→0 f'(x) = Using…

A: Let's find.

Q: Зу Let flx,y) = , find the lim(xy)¬(-1,1)( Зу %3D x+y' ) along the curve y=x² .

A:

Q: Evaluate the following limits.

A:

Q: Find the limit of the function along the path y = 2x X-1 lim (x.y)-(0,0) y -1

A: We have to find the limit of the function along the path

Q: thát limx,y)→(0,0) x6- dõés not exist by using the two-paths approach.

A:

Q: Find the equation of the tangent to the curve y = 2x² – 1 at the point where z =1. Use the limit…

A: We know the standard form of tangent line is     ---&gt;     (y-y0) = m (x-x0)   where 'm' is the…

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.  

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.