# Calculus : Velocity, Continuity, Limits, Differentiability and Integrability

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26) The vertical height in feet of a ball thrown upward from a cliff is given by s(t)=-16t^2+64t+200, where t is measured in seconds. What is the height of the ball, in feet, when its velocity is zero?

27) If the function f is continuous for all real numbers and the limit as h approaches 0 of f(a+h)-f(a)/h = 7 then which statement is true? Please tell why each case is right or wrong and give answer.

a) f(a)=7

b) f is differentialbe at x=a.

c) f is differentiable for all real numbers.

d) f is increasing for x>0.

e) f is increasing for all real differentiable.

25) If f is continuous for all x, which have the same value? Explain.

I. Integral with b on top and a on bottom f(x)dx.

II. Integral with b-a on top and 0 on bottom f(x+a)dx.

III. Integral with b+c on top and a+c on bottom f(x+c)dx.

Answer is I and II only. Tell me why and tell why I is not right.

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Problems involving velocity, continuity, limits, differentiability and integrability are solved and explained in detail.

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26) The verticle height in feet of a ball thrown upward from a cliff is given by s(t)=-16t^2+64t+200, where t is measured in seconds. What is the height of the ball, in feet, when its velocity is zero?

Solution. We first find its velocity, denoted by v(t). Then

So, let .

So,

27)If the function f is continuous for all real ...

###### Education

- BSc , Wuhan Univ. China
- MA, Shandong Univ.

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- "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
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