Tangent Gradient at y-Intercept (HSC SSCE Mathematics Advanced): Quizzes

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Tangent Gradient at y-Intercept
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What is the key difference between a tangent and a secant on a curve?

Tangent touches at 1 pt; secant at 2 pts

For exponential functions y=axy = a^x, where is the y-intercept always located?

At (0,1)(0, 1)

Which formula estimates the tangent gradient to y=axy = a^x at x=0x = 0 using a secant?

m=ah1hm = \frac{a^h - 1}{h}

What happens to the secant approximation as hh approaches zero?

It becomes a better approximation

For y=2xy = 2^x, how does the tangent gradient at (0,1)(0, 1) compare to 1?

Less than 1

Using m=30.0110.01m = \frac{3^{0.01} - 1}{0.01}, what is the gradient (to 4 d.p.)?

1.1047

In the formula m=ah1hm = \frac{a^h - 1}{h}, what does hh represent?

Horizontal distance between two points

How many points on a curve does a tangent line touch?

Exactly one point

Why is the y-intercept of y=axy = a^x always 1 for any positive a1a \neq 1?

Because a0=1a^0 = 1 for any positive aa

What does a positive gradient indicate about a line?

It slopes upward from left to right

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