3 sin x if x + 0 f(x) = a if x = 0.

Q: f(«) = { Making a Function Continuous Find all values of c such that f is continuous on (-0, 0). [1…

A: Given: fx=1-x2,     x≤c fx=x,            x>c fx is continuous on -∞, ∞ Since, fx is continuous…

Q: 7. Derivative of the Reciprocal Use the limit definition to prove f'(x) 5²(x) d 7. dx (f(x), Hint:…

A: To prove that: The derivative of a function is given by in terms of limit h tends to zero is given…

Q: tangent equation at given value points. F(x+h)-f(x) Use f'(x) = lim h-0 b. w = g(z) = 1+ 4-z, (z,w)…

A: given function is w=g(z)=1+4-z find the equation of  the tangent at (3,2) find the differentiation…

Q: se the second-derivative test to determine the local extrema cof the function f. If the…

A:

Q: QI) The function f (x) called invariant under differentiation opcrator function (IUDO. function) if:…

A: Proof is given in step-2

Q: f(x+h)-f(x) Given a function f(x),the derivative is given by f(x) = lim h-0 Let f(x)=V 3x-2 be a…

A: The derivative of a function is defined as, f'x=limh→0fx+h-fxh The function is given as, fx=3x-2 It…

Q: Continuous extension Find a value of c that makes the function 9x - 3 sin 3x 5x x + 0 f(x) = x = 0…

A: is continuous at x=0 We need to find the value of c.

Q: What is the First Derivative Test used for? Testing where a continuous function is differentiable. O…

A:

Q: Using a method show in class, find the intervals on which fis increasing and the intervals on which…

A: Solution:  Increasing and decreasing of a function

Q: Formula for slope: f(x + h) – f(x) m = lim h→0 h 1. Given the function: f(x) = x² + 2x + 1 where x =…

A: The given function f(x)=x2+2x+1 , x=3m=limh→0f(x+h)-f(x)h

Q: 7. (Graphically -Vertical Tangent Line!) The function f (x) = x1/3 is continuous at x = 0, however…

A:

Q: Let f(x) = 2/x the four- step process for finding f^ prime (x)=lim h 0 f(x+h)-f(x) h find the…

A:

Q: Find the absolute extrema of the function on the closed interval. Use a graphing utility to verify…

A: Given function fx=x3-3x First find derivative of given function to find critical points…

Q: IV. Exercišēs Determine if the following functions are continuous at the given value of x. 3. f(x) =…

A: check the value of function for finding whether its continuous or not.

Q: Continuous Function Find a value for a so that the function - 1, x<3 [x² f(x) = \ 2ax,

A: fx=x2-1,     x&lt;32ax,        x≥3 Since the given function is a polynomial function for all x&lt;3…

Q: Identifying discontinuities Determine whether the following functionsare continuous at a. Justify…

A: Check list to determine for given  function is  continuous.    1)…

Q: using Contiaiui the Continuity af the function: at the Conditions of 2 Yょ? 12

A: Introduction: To be continuous at a point, a function must be defined there. Its limit must exist…

Q: Tangent Line. If p(x) = 2x+ 6x + +2, (a) Then p'(1) = !! (b) The equation of the tangent line to the…

A:

Q: lanarithmic differentiation bo find desivative of the function. (x+)"(x-s y= (x-3).

A:

Q: Generalized Mean-Value Theorem: Let f and g be two functions, each having a derivative (finite or…

A: This is a problem of Cauchy MVT.

Q: Slope of a tangent line Let f(x) = 2x 3 - 15x 2 + 24x. For what values of x does the line tangent to…

A: Here,f(x)=2x3-15x2+24x.Now,f'(x)=ddx(2x3-15x2+24x)       =6x2-36x+24

Q: Product Rule for three functions Assume that f, g, and h are dif-ferentiable at x.a. Use the Product…

A: Let us assume that f, g, and h are differentiable at x.   We have to use the Product Rule (twice)…

Q: Making a Function Continuous Find all values of the constant a such that f is continuous for all…

A: fx=axtanx,   x≥0a2-2,  x&lt;0 Given function is continuous for all x&lt;0  &amp;  x&gt;0. Let us…

Q: Theoretical assignment Consider hyperbolic tangent function exp(x) – exp(-æ) exp(x)+ exp(-æ) tanh(x)…

A: Linear Spline Intepolation: A linear spline is a sequence of straight lines. So, in between the data…

Q: Explain how the first derivative of a function determines where the function is increasing and…

A: slope m = y2-y1x2-x1 . Here differentiation of function fI(x) is the slope og the function f(x) . if…

Q: Secant and Tangent Lines Suppose f is a continuous function that goes through the following points:…

A:

Q: Find f(x) where we cannot solve for f'(x) = 0, but f(x) cannot be = 1/x

A: Division rule of derivatives: uv'=vu'-uv'v2  Let us take f(x)=1x2 Solve for f'(x)=0

Q: Continuity of a piecewise function Let For what values of a is g continuous?

A: Given: gx=1-cosx2xif x≠0aif x=0 We know that a function is continuous at x=c if: limx→cfx=fc…

Q: 28. Second-Derivative Test for Concavity Suppose that f is twice differentiable on an open interval…

A: Given: The f is twice differentiable on an open interval I.

Q: Continuity At what points of ℝ2 are the following functions continuous? g(x, y) = ln (x - y)

A: Given function: gx,y=lnx-y Here the  function is the composition fpx,y Where, the value of…

Q: Inconclusive tests Show that the Second Derivative Test is inconclusive when applied to the…

A: To show second derivative test is inconclusive.

Q: Mean Value Theorem. So I determined that this is continuous and differentiable at [1,7]. I just do…

A: Given function is  fx=1x

Q: The Generalized Product Rule Let u, v and w be differentiable function in x. (a) State the product…

A: Note: Since, you have posted a question with multiple sub-parts, we will solve first three subparts…

Q: Continuity. Find out, and give reason, whether f(z) is continuous at z = 0 if f(0) = 0 and for z #0…

A:

Q: Integrating derivatives Use integration by parts to show that if f' is continuous on [a, b], then…

A: Given that f' is continuous on [a, b]. Let us consider: Let f(x) be the first function and f'(x) be…

Q: Consider the function on the interval (0, 27). x)8 sin(x) 8 sin(x) (a) Find the open interval(s) on…

A: Given function is

Q: (a) Let f be differential. Define a function f by the equation f*(x) = lim f(x+Ax)-f(x-Ax) Ax-0 Ax…

A:

Q: EVALUATION Directions: Find the derivative of the following functions. Choose the best answer from…

A:

Q: Graph of f(x) Graph of / A 1. In the interval 0<x< 1 of the first derivative test, f(x) is Ⓒa.…

A:

Q: Continuity of compositions Find functions ƒ and g such that each function is continuous at 0 but ƒ ∘…

A: Find functions f(x) and g(x) which are continuous at x=0 but f(g(x)) is discontinuous at the same.…

Q: (Fixed-point property) Show that if f is continuous on [0, 1] and 0 < f(x)<1 for all x e [0, 1],…

A: g(x) = x - f(x)  And by using intermediate value theorem,

Q: За? — 6х + 7 - g(x) x2 + x + 1

A:

Q: sin(x) The function f(x) has an absolute maximum on the interval [–2, 2].

A: this is false

Q: Numerical Methods Find the numerical second derivative value of f ''(0.5) from the dataset using the…

A: Introduction: When the function values are given in tabular form instead of explicit definition, we…

Q: 2. If a function f(x) is differentiable at x-a, then function f(x) is continuous at? А. х-а В. х-х…

A: Given problems:-

Q: Limits of functions Evaluate the following limits.

A: limx, y→0, πcosxy+sinxy2y

Q: Derivatives Involving Trigonometric Functions 3x² tan x Let f(x) = Find f"(x)and f' (-) sec x

A: We will solve the problem

Q: Use differential calculus to find the exact intervals where the function is increasing or…

A: Given that: f(x)=x4-4x3-8x2+5

Q: Continuity at a point Determine whether the following functions are continuous at a. Justify each…

A: a) Given: f1x2=3x2+2x+1x-1 at the point a=1,  f1=312+21+11-1      =70 Therefore, f1x2 is not…

Q: The Second Derivative Test Use the Second Derivative Test to locatethe local extrema of the…

A: Given function is  (A) fx=3x4-4x3-6x2+12x+1  and x∈-2, 2 Differentiate the fx with respect to x.…

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.