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What is  at x = 1 equal to?

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More Applications of Derivatives Questions

Q1.The growth model for a population is Nt = N0ert where Nt is the population at time t, N0 is the initial population and r is the per capita growth rate. How long does it take for the population numbers to double?
Q2.Suppose that the growth rate of the number of individuals (N) in a population in an environment with a carrying capacity K is modelled as \(\frac{d N}{d t}=r N \frac{K-N}{K}\) where r is the per capita growth rate and t is time; r and K are positive constants. At which of the following values of N will the population growth rate be the maximum?
Q3.The population P = P(t) at time ‘t’ of a certain species follows the differential equation dP/dt = 0.5P – 450. If P(0) = 850, then the time at which population becomes zero is:
Q4.If the tangent to the curve y = x3 at the point P(t, t3) meets the curve again at Q, then the ordinate of the point which divides PQ internally in the ratio 1 ∶ 2 is:
Q5.The function f(x) = (4x3 – 3x2)/6 – 2sin x + (2x – 1)cos x:
Q6.Let the function, f : [-7, 0] → R be continuous on [−7, 0] and differentiable on (−7, 0). If f(-7)= −3 and f'(𝑥) ≤ 2, for all x ∈ (−7, 0), then for all such functions f, f(−1) + f(0) lies in the interval:
Q7.Let f(x) be a differentiable function defined on [0,2] such that f’(x) =f ‘(2 – x) for all x∈(0, 2), f(0) = 1 and f(2) = e2. Then the value of \(\int_0^2 f(x) d x\) is:
Q8.For which of the following curves, the line x + √3y = 2√3 is the tangent at the point (3√3/2, 1/2)?
Q9.If the curve y = ax2 + bx + c, x ∈ R passes through the point (1, 2) and the tangent line to this curve at origin is y = x, then the possible values of a, b, c are:
Q10.Let f be a twice differentiable function defined on R such that f(0) = 1, f'(0) = 2 and f'(x) ≠ 0 for all x∈R. If |f(x) f'(x) f'(x) f”(x)| = 0, for all x∈R, then the value of f(1) lies in the interval:

More Differential Calculus Questions

Q1.If log 2 = 0.2614, log 3 = 0.3521, log 6 = ?
Q2.The growth model for a population is Nt = N0ert where Nt is the population at time t, N0 is the initial population and r is the per capita growth rate. How long does it take for the population numbers to double?
Q3.Suppose that the growth rate of the number of individuals (N) in a population in an environment with a carrying capacity K is modelled as \(\frac{d N}{d t}=r N \frac{K-N}{K}\) where r is the per capita growth rate and t is time; r and K are positive constants. At which of the following values of N will the population growth rate be the maximum?
Q4.The population P = P(t) at time ‘t’ of a certain species follows the differential equation dP/dt = 0.5P – 450. If P(0) = 850, then the time at which population becomes zero is:
Q5.If the tangent to the curve y = x3 at the point P(t, t3) meets the curve again at Q, then the ordinate of the point which divides PQ internally in the ratio 1 ∶ 2 is:
Q6.The function f(x) = (4x3 – 3x2)/6 – 2sin x + (2x – 1)cos x:
Q7.Let the function, f : [-7, 0] → R be continuous on [−7, 0] and differentiable on (−7, 0). If f(-7)= −3 and f'(𝑥) ≤ 2, for all x ∈ (−7, 0), then for all such functions f, f(−1) + f(0) lies in the interval:
Q8.Let xk + yk = ak, (a, k > 0) and \(\frac{d y}{d x}+\left(\frac{y}{x}\right)^{\frac{1}{3}}=0\), then k is
Q9.If \(y(\alpha)=\sqrt{2\left(\frac{\tan \alpha+\cot \alpha}{1+\tan ^2 \alpha}\right)+\frac{1}{\sin ^2 \alpha}}\) where \(\alpha \in\left(\frac{3 \pi}{4}, \pi\right)\), then \(\frac{d y}{d \alpha}\ \rm at\ \alpha=\frac{5 \pi}{6}\) is
Q10.If f(a + b + 1 – x) = f(x) ∀x, where a and b are fixed positive real numbers, then \(\rm \frac{1}{(a+b)} \int_a^b x(f(x)+f(x+1)) d x\) is equal to
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Q1.log10 10000 = _______?
Q2.Consider a system equation \(2x+y-z=0\), \(4x - py + 4z = 4\) and \(x-y+z=q\) where \(p, q \in I\) and \(p, q \in [1, 10]\), then identify the correct statement(s).   List-I List-II (I) Number of ordered pairs \((p, q)\) for which system of equation has unique solution is (P) 1 (II) Number of ordered pairs \((p, q)\) for which system of equation has no solution is (Q) 9 (III) Number of ordered pairs \((p, q)\) for which system of equation has infinite solution is (R) 91 (IV) Number of ordered pairs \((p, q)\) for which system of equation has atleast one solution is (S) 90
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Q7.Suppose that the growth rate of the number of individuals (N) in a population in an environment with a carrying capacity K is modelled as \(\frac{d N}{d t}=r N \frac{K-N}{K}\) where r is the per capita growth rate and t is time; r and K are positive constants. At which of the following values of N will the population growth rate be the maximum?
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