1) Let f(x) be a continuous, everywhere differentiable function and g(x) be its derivative. If f(c)...

1) Let f(x) be a continuous, everywhere differentiable function and g(x) be its derivative. If f(c) = n and g(c) = d, write the equation of the tangent line at x = c using only the variables y, x, c, n, and d. You may use point-slope or slope-intercept but do not introduce more variables.

2) Let f(x) be a continuous, everywhere differentiable function. What kind information does f'(x) provide regarding f(x)?

3) Let f(x) be a continuous, everywhere differentiable function. What kind information does f''(x) provide regarding f(x)?

4) Let f(x) be a continuous, everywhere differentiable function. What kind information does f''(x) provide regarding f'(x)?

5) Let h(x) be a continuous function such that h(a) = m and h'(a) = 0. Is there enough evidence to conclude the point (a, m) must be a maximum or a minimum? Explain.

Expert Solution

milcah answered 2 years ago

Suppose that f is a differentiable function with derivative f" (x) = (x − 3)(x +...

Suppose that f is a differentiable function with derivative f" (x) = (x − 3)(x + 1)(x + 5). Determine the intervals of x for which the function of f is increasing and decreasing Explain why a positive value for f" (x) means the graph f(x) is increasing For f(x) = 2x2- − 3x2 − 12x + 21, find where f'(x) = 0, and the intervals on which the function increases and decreases Determine the values of a, b, and...

Let f be a continuous function on [a, b] which is differentiable on (a,b). Then f...

Let f be a continuous function on [a, b] which is differentiable on (a,b). Then f is non-decreasing on [a,b] if and only if f′(x) ≥ 0 for all x ∈ (a,b), while if f is non-increasing on [a,b] if and only if f′(x) ≤ 0 for all x ∈ (a, b). can you please prove this theorem? thank you!

Let f be a differentiable function on the interval [0, 2π] with derivative f' . Show...

Let f be a differentiable function on the interval [0, 2π] with derivative f' . Show that there exists a point c ∈ (0, 2π) such that cos(c)f(c) + sin(c)f'(c) = 2 sin(c).

Suppose f is a twice differentiable function such that f′(x)>0 and f′′(x)<0 everywhere, and consider the...

Suppose f is a twice differentiable function such that f′(x)>0 and f′′(x)<0 everywhere, and consider the following data table. x 0 1 2 f(x) 3 A B For each part below, determine whether the given values of A and B are possible (i.e., consistent with the information about f′and f′′ given above) or impossible, and explain your answer. a)A= 5, B= 6 (b)A= 5, B= 8 (c)A= 6, B= 6 (d)A= 6, B= 6.1 (e)A= 6, B= 9

prove using the definition of derivative that if f(x) and g(x) is differentiable than (f'(x)g(x) -...

prove using the definition of derivative that if f(x) and g(x) is differentiable than (f'(x)g(x) - f(x)g'(x))/g^2(x)

Rolle's Theorem, "Let f be a continuous function on [a,b] that is differentiable on (a,b) and...

Rolle's Theorem, "Let f be a continuous function on [a,b] that is differentiable on (a,b) and such that f(a)=f(b). Then there exists at least one point c on (a,b) such that f'(c)=0." Rolle's Theorem requires three conditions be satisified. (a) What are these three conditions? (b) Find three functions that satisfy exactly two of these three conditions, but for which the conclusion of Rolle's theorem does not follow, i.e., there is no point c in (a,b) such that f'(c)=0. Each...

Part A. If a function f has a derivative at x not. then f is continuous...

Part A. If a function f has a derivative at x not. then f is continuous at x not. (How do you get the converse?) Part B. 1) There exist an arbitrary x for all y (x+y=0). Is false but why? 2) For all x there exists a unique y (y=x^2) Is true but why? 3) For all x there exist a unique y (y^2=x) Is true but why?

Question 1 Let f be a function for which the first derivative is f ' (x)...

Question 1 Let f be a function for which the first derivative is f ' (x) = 2x 2 - 5 / x2 for x > 0, f(1) = 7 and f(5) = 11. Show work for all question. a). Show that f satisfies the hypotheses of the Mean Value Theorem on [1, 5] b)Find the value(s) of c on (1, 5) that satisfyies the conclusion of the Mean Value Theorem. Question 2 Let f(x) = x 3 − 3x...

For the following exercises, determine whether or not the given function f is continuous everywhere...f(x) = log2 (x)

For the following exercises, determine whether or not the given function f is continuous everywhere. If it is continuous everywhere it is defined, state for what range it is continuous. If it is discontinuous, state where it is discontinuous.f(x) = log2 (x)

Suppose f(x) is a very well behaved function in that it is continuous and differential everywhere,...

Suppose f(x) is a very well behaved function in that it is continuous and differential everywhere, and that f(2) = 6 and f(6) = 34. a. Find the slope of the line between these two points on the graph. b. What does the Mean Value Theorem tell you about f(x) in the interval between x = 2, and x = 6?

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